MATH BLASTER GEOMETRY - Knowledge...

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MMAATHTH BBLASTERLASTERGGEOMETREOMETRYY ™™

TTEACHEREACHERMMAATERIALSTERIALS

0361601


PROGRAM OVERVIEW . . . . . . 5

INTRODUCTION . . . . . . . . . . . . 6

PART 1: GEOMETRY TOPICS

UNIT 1: POINTS, LINES, PLANES,AND ANGLES. . . . . . . . . . . . . . . . . 7

The Whole TruthName That AngleTrue or False?Label the DiagramFind Your Own Angles

UNIT 2: TRIANGLES . . . . . . . . . . . . 16Characteristics of a TriangleTriangle ApplicationsCongruent TrianglesFind the Fastest RoutePoints of Concurrency

UNIT 3: QUADRILATERALS AND

OTHER POLYGONS . . . . . . . . . . . 26Tree DiagramQuadrilatral RelationshipsConcave or Convex?Cut Out the AnglesAngles of Polygons

UNIT 4: SIMILARITY . . . . . . . . . . 35Ratio and ProportionExploring SimilaritySimilar PolygonsSpecial Right TrianglesConstructing Similar Polygons

UNIT 5: CIRCLES . . . . . . . . . . . . . 45Circle ConstructionMore Circle ConstructionParts of a CircleLabel the CircleApplications with Circles

MATH BLASTER GEOMETRY™

TABLE OF CONTENTS


TABLE OF CONTENTS

CONTINUED

UNIT 6: PERIMETER AND AREA . . 54Area of ParallelogramsArea of TrianglesArea of TrapezoidsArea of CirclesRecreation Area

UNIT 7: SOLIDS . . . . . . . . . . . . . . 64Characteristics of SolidsSurface Area of SolidsVolumeVolume and Surface Area Applications

UNIT 8: COORDINATE GEOMETRY . . . 73Distance FormulaMidpoint FormulaQuadrilaterals on Coordinate GridsMore Quadrilaterals on Coordinate GridsEquation of a Line

UNIT 9: TRANSFORMATIONAL

GEOMETRY. . . . . . . . . . . . . 83Translations and RotationsReflectionsTransformationsTranslations and Reflections on a

Coordinate PlaneRotations on a Coordinate Plane

UNIT 10: REASONING AND

PROOF . . . . . . . . . . . . . . . . 93Logical ReasoningCut-and-Paste Proofs

PART 2: BRAIN TEASERS. . . 102GeoboardTangramsGeometry Blaster Game ConnectionsConstructionsWorld Wide Web Sites

4 Math Blaster® Geometry

Reproduction of these pages by the classroom teacher for use in the classroom is permissible. The reproduction ofany part of this book for an entire school or school system or for commercial use is strictly prohibited

© Vivendi Universal Publishing and/or its subsidiaries. All Rights Reserved. Math Blaster is a registered trademark of Vivendi Universal Publishing and/or its subsidiaries.

All trademarks referenced herein are the property of their respective owners.

CONTENT DEVELOPERSPAM HALLORAN

KATHLEEN S. COLEMAN

MICHELE LEONETTI

ALISON BIRCH

DESIGN AND LAYOUTLAURIE GALVAN

PAM WISSINGER

EDITORJOE SKELLEY


MATH BLASTER GEOMETRY™

MATH BLASTERGEOMETRY™

LEVELHigh School

ACTIVITY COMPONENTSThe evil Geometrons are downsizing Zoid’s home planetfrom 3-D to 2-D! Learn geometry principles as you collectpieces to restore the Dimension Machine.

Building of Truth – Help Andi and Zoid obtain apiece by classifying geometric figures.

Pit of Despair – Adjust your angle to shoot targets toremove the shield covering the puzzle piece.

Proof Palace – Rearrange the statements or reasonsand solve the proofs.

N-Gon Mountains – Scale the highest peak byanswering multiple-choice questions.

Capitol Building – Match up the pairs based on therule that is known.

Sphinx – Answer the question to unlock the door to theDimension Machine.

Dimension Machine – What would each piece looklike if folded into a cube? Answer this to repair themachine.

FEATURES• Animated lessons make

geometry come alive• Follows National Council of

Teachers of Mathematics guide-lines

• Content connects geometric prin-ciples to real-life applications

• Friendly interface enables quickaccess to specific subjects andlevels

• On-screen help and glossary of geometry terms

• Records and tracks students‘progress

SKILL DEVELOPMENT

• Inductive, deductive, and geometric proofs

• Attributes of triangles, circles, and quadrilaterals

• Symbols, formulas, postulates and theorems

• Similarity with ratios and proportions

• Area and volume of solids – prisms, pyramids, cones,cylinders and spheres

• Compass and protractor use

• Coordinate and transformational geometry


INTRODUCTION

Dear Colleague,

Welcome to Math Blaster Geometry!Our primary goal is to help youteach geometry in a way thatengages your students’ interest,interactively enabling them to under-stand and appreciate the place ofgeometry in mathematics class, theenvironment, or even a potentialcareer field.

By including a full year of geometrytopics, we help streamline yourteaching preparations. The 52 sub-ject areas in the program, combinedwith these print activities, encourageand enable your students to under-stand and apply geometric conceptsin algebra, trigonometry, and tradi-tional proofs.

Math Blaster Geometry offers customized assessment and individualized instruction that increase efficiency and allow more timefor individualized instruction and planning.

We invite your comments and obser-vations. Please send them to us by mailor at our Internet address (http://www.education.com). We lookforward to hearing from you.

Yours in mathematics education,

The Blaster DevelopmentGroup


UNIT 1 – POINTS, LINES, PLANES, AND

ANGLES

Lesson 1: The Whole Truth

Lesson 2: Name That Angle

Lesson 3: True or False?

Lesson 4: Label the Diagram

Lesson 5: Find Your Own Angles


LESSON 1: THE WHOLE TRUTH

Hand out copies of the Whole Truth activitysheet, p. 9. This activity includes 15 statementsabout different characteristics of points, lines, andplanes. The information in each statement is eitheralways true, sometimes true, or never true. As stu-dents read the statements, they are encouraged toconsider different situations with lines, points, andplanes before writing whether the statement isalways, sometimes, or never true. Once studentshave completed the activity, you may wish to dis-cuss students’ reasons for the answers they wrote.

LESSON 2: NAME THAT ANGLE

Hand out copies of the Name That Angleactivity sheet, p. 10. To define each of the termson this page, students must understand the char-acteristics of several kinds of angles, bisectors,and lines. Students then apply the definitions bydrawing an example for each term. Remind stu-dents to label their drawings. Some students mayprefer to draw only one or two diagrams whichinclude examples of all the terms.

LESSON 3: TRUE OR FALSE?

Hand out copies of the True or False? activitysheet, p. 11. In this activity, students go beyondnaming characteristics and definitions. Each state-ment describes a situation involving angles, bisec-tors, or lines that may or may not be true. Youmay want to suggest that students make drawingsto help them decide if a statement is true or false.Students can compare answers and discuss whythe statements can or cannot be true.

LESSON 4: LABEL THE DIAGRAM

Hand out copies of the Label the Diagramactivity sheet, p. 12. This lesson provides studentsthe opportunity to apply the characteristics ofangles and bisectors by labeling a diagram. Thestatements given lead to conclusions about thelocations of points and the measures of angles.Point out that the steps should be performedsequentially.

Instruct students to keep the completed activitysheet. They will need the diagram for the exercis-es in Lesson 5.

LESSON 5: FIND YOUR OWNANGLES

Hand out copies of the Find Your OwnAngles activity sheet, p. 13, and have studentstake out their copies of Lesson 4. In Part A ofthe new lesson, students find examples of differentkinds of lines, angles, and bisectors. This involvesstudying and analyzing all aspects of the dia-gram. You may find it helpful to review propernotation for angles.

In Part B, students are given a pair of angles and then asked to classify these angles in twodifferent ways. When students explain theiranswers, they clarify their thinking and demon-strate understanding.

UNIT 1 – POINTS, LINES, PLANES, AND ANGLES


UNIT 1 LESSON 1

THE WHOLE TRUTH NAME __________________________

Directions: Read each statement carefully. Determine if the statement is always true,sometimes true, or never true. Complete each sentence with the appropriate word:always, sometimes, or never.

1. A line is _______________________ labeled by a single lowercase letter.

2. If two distinct lines intersect, they _____________________ intersect at exactlyone point.

3. If two planes intersect, they _______________________ intersect at exactlyone point.

4. Two collinear points are _______________________ coplanar.

5. Two coplanar points are _______________________ collinear.

6. Three coplanar points are _______________________ collinear.

7. Three collinear points are _______________________ coplanar.

8. Given two points A and B in a plane, ray AB and ray BA are_______________________ the same ray.

9. The intersection of a line and a plane is _______________________ one point.

10. A point _________________ has size, but __________________ has a location.

11. A segment is _______________________ part of a line.

12. AB, AB, and AB _______________________ refer to the same set of points.

13. If two segments are coplanar, they _______________________ intersect ata point.

14. A plane _______________________ extends in three dimensions.

15. Two distinct points _______________________ determine exactly one line.

UNIT 1 LESSON 2

NAME THAT ANGLE NAME __________________________

Directions: Define each term and draw an example.

A. Acute angle:

B. Obtuse angle:

C. Straight angle:

D. Adjacent angles:

E. Complementary angles:

F. Supplementary angles:

G. Vertical angles:

H. Segment bisector:

I. Angle bisector:

J. Perpendicular lines:



UNIT 1 LESSON 3

TRUE OR FALSE? NAME __________________________

Directions: For each statement, write T (true) or F (false).

1. Both angles in a pair of complementary angles can be obtuse. ___________

2. Two perpendicular lines create congruent, adjacent angles. ___________

3. Two vertical angles always have a common vertex. ___________

4. A segment bisector creates two congruent segments. ___________

5. Two complementary angles always have a common vertex. ___________

6. Two vertical angles can be obtuse. ___________

7. Each point on an angle bisector is equidistant from the two sides. ___________

8. Both angles in a pair of supplementary angles can be obtuse. ___________

9. The angles of a pair of supplementary angles can be acute or obtuse. ___________

10. An angle bisector can be a point. ___________

11. Two angles complementary to the same angle are congruent to each other. _________

12. Both angles of a pair of supplementary angles can be acute. ___________

13. Two perpendicular lines always form four right angles. ___________

14. A pair of vertical angles are always congruent. ___________

15. By definition, two complementary angles must also be adjacent angles.___________

16. A segment bisector can be a segment, ray, or line. ___________

17. By definition, two supplementary angles must also be adjacent angles. ___________

18. Together, a pair of adjacent supplementary angles form a straight angle. __________


UNIT 1 LESSON 4

LABEL THE DIAGRAM NAME __________________________

Directions: Use the information below to complete the diagram. You will label 5 points and 4 angle measures.

Information What to Draw

1. CD bisects AB at point M. 1. Label point M.

2. The measure of �AME is 20°. 2. Label point E.

3. The measure of �BMF is 45°. 3. Label point F.

4. MF bisects �BMD. 4. Label the measure of �DMF.

5. CD is ⊥ AB. 5. Label the 3 missing angle measures.

6. �AMG and �BMH are acute, vertical 6. Label points G and H.angles.

C

A

D

B20˚30˚ 45˚

30˚


UNIT 1 LESSON 5

FIND YOUR OWN ANGLES NAME __________________________

A. Directions: Refer to your completed diagram in Lesson 4. Use proper notation toidentify an example of each of the following items. There is more than one possibleanswer for many of the items.

1. midpoint ____________ 7. angle bisector ____________

2. segment bisector ____________ 8. straight angle ____________

3. acute angle ____________ 9. obtuse angle ____________

4. vertical angles ____________ 10. adjacent angles ____________

____________ ____________

5. complementary ____________ 11. supplementary ____________angles angles

____________ ____________

6. right angle ____________ 12. pair of perpendicular ____________lines

B. Directions: Refer to your completed diagram from Lesson 4. Write two differentangle classifications that apply to each item below. On the back, explain youranswers.

1. �AMC and �BMC __________________ __________________

2. �AMG and �BMH __________________ __________________

3. �DMF and �BMF __________________ __________________

4. �CMG and �DMH __________________ __________________


UNIT 1 – ANSWER KEY

LESSON 1

1. sometimes 9. sometimes2. always 10. never, always3. never 11. always4. always 12. never5. always 13. sometimes6. sometimes 14. never7. always 15. always8. never

LESSON 2

Student drawings will vary.A. Acute angle: an angle whose measure is less

than 90°.

B. Obtuse angle: an angle whose measure isgreater than 90°.

C. Straight angle: an angle whose measureequals 180°.

D. Adjacent angles: two angles in a plane witha common vertex and a common side, but nocommon interior points.

E. Complementary angles: two angles whosemeasures have a sum of 90°.

F. Supplementary angles: two angles whosemeasures have a sum of 180°.

G. Vertical angles: two opposite angles formedby intersecting lines.

H. Segment bisector: a segment, ray, or line thatdivides a segment into two equal parts.

I. Angle bisector: a ray that divides an angleinto two congruent angles.

J. Perpendicular lines: two lines that intersect toform right angles.

LESSON 3

1. F 10. F2. T 11. T3. T 12. F4. T 13. T5. F 14. T6. T 15. F7. T 16. T8. F 17. F9. T 18. T

C

M

A

D

B20˚

70˚60˚

H

F

G

E

60˚ 45˚30˚ 45˚

30˚

LESSON 4



LESSON 5

Answers may vary. Sample answers are provided.

A. 1. M2. CD3. �EMA4. �GMD and �CMH5. �CMH and �BMH6. �AMD7. MF8. �AMB9. �EMH

10. �DMG and �DMF11. �AMF and �BMF12. CD ⊥ AB

B.1. �AMC and �BMC are right angles, since

they are formed by two perpendicular lines.They are also adjacent angles because theyshare a vertex and side, but don’t share interior points.

2. �AMG and �BMH are both acute anglesbecause their measures are less than 90°.They are also vertical angles because theyhave a common vertex and are oppositeangles formed by the intersecting lines HGand AB.

3. �DMF and �BMF are complementaryangles because the sum of their measuresequals 90°. They are also adjacent anglessince they share a vertex and a side but don’t share interior points.

4. �CMG and �DMH are vertical angles be-cause they have a common vertex and areopposite angles formed by the intersectinglines CD and GH. Both angles are obtusesince their measures are greater than 90°.


UNIT 2 –TRIANGLES

Lesson 1: Characteristics of a Triangle

Lesson 2: Triangle Applications

Lesson 3: Congruent Triangles

Lesson 4: Find the Fastest Route

Lesson 5: Points of Concurrency


LESSON 1: CHARACTERISTICS OF ATRIANGLE

Hand out copies of the Characteristics of aTriangle activity sheet, p. 18. Students should befamiliar with the characteristics of different kindsof triangles. In Part A, they use the informationprovided on the triangle drawings to determine allpossible labels for each figure. Suggest that stu-dents study the relationship of the sides of the tri-angles as well as the angle measures.

For Part B, make sure students understand thatthey can draw any triangles that fit the givendescriptions. Tell students that their drawings canbe labeled with angle measures, side measures,congruent symbols, or right angle symbols.

LESSON 2: TRIANGLE APPLICATIONS

Hand out copies of the Triangle Applicationsactivity sheet, p. 19. You may wish to assign thislesson at or near the end of the unit. Students ap-ply a variety of information from throughout theunit to find the missing measures of the triangles.Allow students to use whatever strategies theywant in order to solve the problems. You may wishto have students compare methods once they havecompleted the activity.

LESSON 3: CONGRUENT TRIANGLES

Hand out copies of the Congruent Trianglesactivity sheet, p. 20. This activity requires studentsto apply the different congruency theorems presented in this unit in determining congruent triangles. Ask students which theorems they used to solve the problems, and then ask volun-teers to present different ways of solving the same problems.

LESSON 4: FIND THE FASTEST ROUTE

Hand out copies of the Find the Fastest Routeactivity sheet, p. 21. To complete the activity, stu-dents label distances on the map. One missingdistance can be found by subtracting, while theother distance is determined by applying thePythagorean theorem. Once students know all thedistances, they use the travel speeds given todetermine the approximate time it takes to get tothe airport by each route. Some students mightuse proportions to find the answers. If necessary,review concepts related to ratios and proportions.

You could provide extensions to this problem bychanging the travel times on the roads. You mightchallenge students to write their own problemssimilar to this one.

LESSON 5: POINTS OF CONCURRENCY

Hand out copies of the Points of Concurrencyactivity sheet, p. 22. When students explore con-cepts and make their own discoveries, they arelikely to better understand and remember rules. Inthis activity, students construct angle bisectors andperpendicular bisectors to examine the distancefrom the points of concurrency to parts of the tri-angle. They measure distances and write theirown rules about the incenter and circumcenter ofa triangle. As a challenge, students can study theorthocenter (intersection of all altitudes) and cen-troid (intersection of all medians) of triangles tosee if similar rules can be stated.

UNIT 2 – TRIANGLE

UNIT 2 LESSON 1

CHARACTERISTS OF A TRIANGLE NAME __________________________

A. Directions: Circle all the words that describe each triangle.

1. scalene isosceles equilateralacute right obtuseequiangular



B. Directions: Draw a triangle that fits each description and label the identifyingparts.


3. Acute Scalene1. Right Scalene

2. Obtuse Isosceles 4. Right Isosceles

110˚

40˚

30˚

7 7

7

3

4

5


UNIT 2 LESSON 2

TRIANGLE APPLICATIONS NAME __________________________

Directions: Solve for the missing measures.

1. What is the measure of �BAC + �ABC?

________________________________________

2. What is the measure of �ABE?

________________________________________

3. DB is a median. What is the measure of AB?

________________________________________

4. Given that AB = 10 and DC = 6, find the following measures:

AC = ____________

DB = ____________

AD = ____________

�CAD = _________

A

B 50˚D

50˚ C

A

B

C

5

D

3

CDEA

40˚

40˚

B

100˚

C DA

B

30˚

40˚

110˚

UNIT 2 LESSON 3

CONGRUENT TRIANGLES NAME __________________________

Directions: Complete the following 4 steps for each problem:1. Label the figure based on the given information.2. Determine if any two triangles can be proven congruent.3. Write the congruence statement.4. State the theorem that proves the triangles are congruent.

1. Given: ∆ACE is isosceles with AC � CE;BE and AD are medians.

∆___________ � ∆___________

Theorem: ______________________

2. Given: BE ⊥ � AE; AB � CD.∆___________ � ∆___________

Theorem: ______________________

3. Given: ∆ADC is isosceles with AD � DC;∆AEC is isosceles with AE � EC;DB is a median.

∆___________ � ∆___________

Theorem: ______________________

4. Given: ABCD is a rectangle.

∆___________ � ∆___________

Theorem: ______________________


C

B D

EA

B

C

A E D

D

B CA

E

unit 2-11.eps

A

C

B

D


UNIT 2 LESSON 4

FIND THE FASTEST ROUTE NAME __________________________

Directions: Imagine that you are at your home on the corner of 15th Street and Crosstown Boulevard. You need to get to the airport as quickly as possible. Use the map and the information below to answer the questions. Show your work. Round numbers when appropriate.

1. The distance from yourhome down 15th Streetto Main Street is 8 miles.Label this distance on the map.

2. The distance from the corner of 15th Streetand Main Street to the airport is 22 miles. Labelthis distance on the map.

3. The distance from the intersection of Crosstown Boulevard and Main Street to the airport is 16 miles. Label this distance on the map.

4. What is the distance from the intersection of 15th Street and Main Street to the inter-section of Crosstown Boulevard and Main Street? Label this distance on the map.

________________________________________________________________________

5. What is the distance from your home down Crosstown Boulevard to Main Street? Label this distance on the map.______________________________________________

________________________________________________________________________

6. Traffic on 15th Street is moving at about 45 mph. Traffic on Crosstown Boulevard is moving at about 40 mph. Traffic on Main Street is moving at about 40 mph. Approximately how long will it take to get from your home to the airport using the twodifferent routes? __________________________________________________________

________________________________________________________________________

15th

Stre

et

Cros

stow

nBl

vd.

Main Street Airport

Home


UNIT 2 LESSON 5

POINTS OF CONCURRENCY NAME __________________________

Concurrent lines are two or more lines that intersect at the same point. Thepoint of concurrency is the point at which two or more lines intersect. Triangleshave special points of concurrency which can occur either inside or outside of a triangle.

Directions: Construct the points of concurrency based on the information below.Label each point with a letter. Then describe a special characteristic of each point.

1. The incenter of a triangle is the point at which all the angle bisectors of the triangle intersect. Construct the incenter of Triangle 1 above. Label this point X.

2. Measure the distance from the point of concurrency to the three sides.

X to BC = ________ X to AC = ________ X to AB = ________

3. In your own words, what is a special characteristic of the incenter of a circle?

________________________________________________________________________

4. The circumcenter of a triangle is the point at which all the perpendicular bisectors of the sides of a triangle intersect. Construct the circumcenter of Triangle 2. Label this point Y.

5. Measure the distance from the point of concurrency to the three vertices.

AY = ____________ BY = ____________ CY = __________

6. In your own words, what is a special characteristic of a circumcenter?

B

A

C B

A

CTriangle 1 Triangle 2


LESSON 1A.1. scalene, obtuse2. isosceles, equilateral, acute, equiangular3. scalene, right

B. Drawings may vary. Samples are provided.1.

Right scalene triangle

2.

Obtuse isosceles triangle with obtuse angleand congruent sides labeled as indicated

3.

Acute scalene triangle

4.

Right isosceles triangle

LESSON 2

1. �BAC + �ABC + �BCA = 180° and �BCA = 70°, so �BAC + �ABC = 110°.

2. �ABE + 40° + 40° = 100°, so �ABE = 20°.Students might continue to label the measuresof the angles in the diagram to find the measure of �ABE. See the diagram below.

3. Since ACD is a right triangle, AC2

+ 32

= 52,

so AC2

= 16 and AC = 4. Since DB is themedian, AB = BC. So AB = 2.

4. AC = 10 since ABC is an equilateral triangle.DB = 6, since DC = DB.AD = 8 using the Pythagorean theorem.�CAD = 40°, since 90° + 50° + �CAD = 180°.


513

12

100˚

10

16

8

CDEA

40˚

40˚

B

100˚

unit 2-5answ.eps

20˚

120˚ 60˚ 80˚



LESSON 3Solutions may vary. Sample answers are provided.

1. ∆DC�∆BCETheorem: SAS

2. ∆ABE > ∆CDETheorem: HL

3. ∆AEB � ∆CEB Theorem: SSS

unit 2-8answ.eps

C

B D

EA

B

C

A E D

D

B CA

E

4. ∆ABD � ∆DCATheorem: SSS

unit 2-11answ.eps

A

C

B

D



LESSON 4

1–5. See the labeled map below.

6. Home to intersection of 15th and Main: 45 mi/60 min = 8 mi/x; x �11.

Intersection of 15th and Main to airport:40 mi/60 min = 22 mi/x; x = 33.

This route would take approximately 11 + 33, or 44 minutes.

Home to intersection of Crosstown and Main:40 mi/60 min = 10 mi/x; x = 15.

Intersection of Crosstown and Main to airport: 40 mi/60 min = 16 mi/x; x = 24.

This route would take approximately 15 + 24, or 39 minutes.

LESSON 5

Students’ measurements and constructions mayvary. Students should make conclusions similar tothe following for questions 3 and 6:

3. The incenter of a triangle is equidistant from the sides.

6. The circumcenter of a triangle is equidistantfrom the vertices.

unit 2-12answ.eps

15th

Stre

et

Cros

stow

nBl

vd.

Main Street Airport

8 m

iles

10 m

iles

6 miles 16 miles22 miles

Home


UNIT 3 – QUADRILATERALS AND OTHER

POLYGONS

Lesson 1: Tree Diagram

Lesson 2: Quadrilateral Relationships

Lesson 3: Concave or Convex?

Lesson 4: Cut Out the Angles

Lesson 5: Angles of Polygons


UNIT 3 – QUADRILATERALS AND OTHER POLYGONS

LESSON 1: TREE DIAGRAM

Hand out copies of the Tree Diagram activitysheet, p. 28. Help students understand the formatof the diagram (from general to specific). Thisactivity can serve a variety of purposes. As students complete the diagram, they show anunderstanding of the characteristics of a quadri-lateral. The organization of the diagram makes it easy to study the relationships between quadri-laterals. The completed tree diagram can be used as a visual reference to answer questionsabout the characteristics of all kinds of quadrilat-erals. Have students keep the diagram for usewith Lesson 2.

LESSON 2: QUADRILATERALRELATIONSHIPS

Hand out copies of the QuadrilateralRelation- ships activity sheet, p. 29. Have students also take out their completed TreeDiagram activity sheet from Lesson 1. Explain that the diagram includes all the informa-tion they need to complete the true-or-false exer-cise. Students might draw examples of the differ-ent quadrilaterals on their diagram, including all of the characteristics listed.

LESSON 3: CONCAVE OR CONVEX?

Hand out copies of the Concave or Convex?activity sheet, p. 30. Remind students that a diag-onal of a figure is a line that connects two ver-tices, but is not a side of the figure. This activityinvolves examining the diagonals of polygons todevelop definitions for concave and convex poly-

gons. Once this concept is established, studentsexplore regular and irregular polygons, trying toconstruct concave and convex figures. In all ofthese exercises, students discover characteristicsof polygons and the relationships among thesetraits and figures.

LESSON 4: CUT OUT THE ANGLES

Hand out copies of the Cut Out the Anglesactivi-ty sheet, p. 31. There are a few differentways to study the sum of the measures of anglesin polygons. This technique involves cutting outthe angles of figures and then placing them sideby side on a straight line. The purpose is to lookfor angles that form a straight angle (180°).Students manipulate the pieces and get a hands-on understanding of the concept. During this les-son, you may also wish to take advantage of theopportunity to review angles by looking forexamples of acute, obtuse, right, and supplemen-tary angles.

LESSON 5: ANGLES OF POLYGONS

Hand out copies of the Angles of Polygonsactivity sheet, p. 32. After completing Lesson 4,students should understand what it means to findthe sum of the measures of the angles of a poly-gon. This activity introduces a formula for findingthis sum in polygons with any number of sides.Students find the sums and then solve problemsthat apply the information. The skills they practicehere will be used in proofs and in other problem-solving situations.


UNIT 3 LESSON 1

TREE DIAGRAM NAME __________________________

Under each figure name is one or more lines for listing the characteristics of that figure. Any figure that falls below another on the tree has its own characteristics as well as the characteristics of the figures above it. For example, a rhombus has all of the characteristics of a parallelogram and quadrilateral as well as two addi-tional traits of its own. From top to bottom, the figures in the tree go from general to more specific.

Directions: Choose from the list below. Write the characteristics where they belong in the tree. Each characteristic is used only once.• Diagonals bisect each other• Has 4 right angles• Has exactly 1 pair of parallel sides• Has 2 pairs of parallel sides• Has 4 sides• Opposite angles are congruent

• Has 4 congruent sides• Has 2 pairs of congruent sides• Has exactly 2 congruent legs• All sides are different lengths• Diagonals are perpendicular to

each other

Quadrilateral

Parallelogram

Rhombus Rectangle

Square

Isosceles Scalene

Trapezoid


UNIT 3 LESSON 2

QUADRILATERAL RELATIONSHIPS NAME __________________________

Directions: Use your completed tree diagram from Lesson 1 to help determinewhether each statement below is true or false. Write T or F on the lines.

1. A rectangle is a quadrilateral. ________

2. A rhombus is a rectangle. ________

3. A square is a parallelogram. ________

4. An isosceles trapezoid is a parallelogram. ________

5. A square is a rectangle. ________

6. A rectangle is a square. ________

7. A rectangle is a parallelogram. ________

8. A parallelogram is a rectangle. ________

9. The diagonals of a square bisect each other. ________

10. The diagonals of a rectangle are perpendicular. ________

11. An isosceles trapezoid has two pairs of congruent sides. ________

12. Opposite angles of a rhombus are congruent. ________

13. A parallelogram has four right angles. ________

14. The diagonals of a square are perpendicular bisectors of each other. ________

15. The diagonals of a parallelogram are perpendicular. ________


UNIT 3 LESSON 3

CONCAVE OR CONVEX? NAME __________________________

A. Directions: Draw all the diagonals for each figure below. Then answer the questions.

1. Using the completed drawings, how would you describe a convex polygon?

__________________________________________________________________________

2. Using the completed drawings, how would you describe a concave polygon?

________________________________________________________________________

A regular polygon has congruent sides and angles. If a polygon does nothave congruent sides and congruent angles, it is irregular.

B. Directions: Try to make a sketch of each polygon described below. If the sketchcannot be made, write impossible.

1. a concave, irregular hexagon 3. a concave parallelogram

2. a convex, regular hexagon 4. a concave, irregular quadrilateral

CConcave

ConcaveConvexConvex Concave

Concave


UNIT 3 LESSON 4

CUT OUT THE ANGLES NAME __________________________

Directions: Cut out the quadrilatrals at the bottom of the page. Then follow the directions and answer the questions.

1. Cut off the corners of figure A. Place the cut-off pieces side by side with the angle vertices touching the line below.

2. How many straight angles can you form with the 4 angles of figure A?______________

3. What is the total measure of all of these angles? ________________________________

4. Follow the same procedure with the other figures. What can you conclude about the sum of the measures of the angles of a quadrilateral?____________________________

unit 3-3a.eps

A B

C D


UNIT 3 LESSON 5

ANGLES OF POLYGONS NAME __________________________

Another way to find the sum of the angles of a polygon is by using a formula. This formula is (n – 2) • 180, where n equals the number of sides in the figure.

A. Directions: Use the formula to complete the following:

B. Directions: Use the information in the table to answer the questions.

1. What is the measure of each angle in a regular hexagon? ______________________

2. Seven of the angles in an octagon have a sum of 920°. What is the measure of theeighth angle? ____________________________________________________________

3. What is the measure of each angle in a regular decagon? ______________________

4. Four of the angles of a pentagon have a sum of 432°. Could this figure be a regularpentagon? Why or why not? ______________________________________________

________________________________________________________________________

5. Three of the measures of the angles in a quadrilateral have a sum of 295°. What is the sum of the fourth angle? ________________________________________________

6. Could a heptagon have 7 angles whose measures are equal whole numbers? Why or why not? ________________________________________________________________

________________________________________________________________________

Figure Number of Sides Sum of Measures ofAngles

Quadrilateral ______________ ______________

Pentagon ______________ ______________

Hexagon ______________ ______________

Heptagon ______________ ______________

Octagon ______________ ______________

Decagon ______________ ______________

LESSON 3A.

1. A polygon whose diagonals have points only inside the figure2. A polygon with at least one diagonal that has points outside the figure



Quadrilateral

Parallelogram

Rhombus Rectangle

Square

Isosceles Scalene

Trapezoid

Has 4 sides

Has 2 pairs of parallel sides Has exactly 1 pair of parallel sides

Has 2 pairs of congruent sides

Opposite angles are congruentDiagonals bisect each other Has exactly 2

congruent legsAll sides aredifferent lengths

Has 4 right anglesHas 4 congruent sides

Diagonals are perpendicularto each other

LESSON 1

LESSON 2

1. T2. F3. T4. F5. T

6. F7. T8. F9. T10. F

11. F12. T13. F14. T15. F

Concave

Concave

Convex

Concave

Concave

Convex



LESSON 3, CONT.B. Students’ drawings may vary. Samples are provided.

1.

2.

3. Impossible

4.

LESSON 4

1. Check students’ work.2. 2 straight angles3. 360°4. The sum of the measures of the angles of a

quadrilateral is 360°. Adjacent angles aresupplementary.

LESSON 5

A.Number of Sum of

Figure Sides Measures of AnglesQuadrilateral 4 360°Pentagon 5 540°Hexagon 6 720°Heptagon 7 900°Octagon 8 1080°Decagon 10 1440°

B. 1. 120°2. 160°3. 144°4. Yes. Each of the angles in a regular pentagon

has a measure of 108°, and 108 • 4 = 432.This could be a regular pentagon.

5. 65°6. No. The sum of the measures of the angles of

a heptagon equals 900°, and 900 ÷ 7 would not be a whole-number quotient.

Concave, irregular hexagon

Convex, regular hexagon

Concave, irregular quadrilateral


UNIT 4 – SIMILARITY

Lesson 1: Ratio and Proportion

Lesson 2: Exploring Similarity

Lesson 3: Similar Polygons

Lesson 4: Special Right Triangles

Lesson 5: Constructing Similar Polygons


UNIT 4 – SIMILARITY

LESSON 1: RATIO AND PROPORTION

Hand out copies of the Ratio and Proportionactivity sheet, p. 37. This lesson reviews conceptsthat are important for understanding similarity. Inexercise A, students read about particular situa-tions and then write ratios to describe them. Thedirections do not specify how the ratios should beexpressed or in which order the items shouldappear. For this reason, instruct students to writesome brief descriptive words along with the num-bers, to give more meaning to the ratios.

In exercise B, students write and solve proportionsto answer questions. You may wish to reviewcross products. Remind students to be careful thatthe numerators and denominators in their equiva-lent ratios are comparing the appropriate items.

LESSON 2: EXPLORING SIMILARITY

Hand out copies of the Exploring Similarityactivity sheet, p. 38. When students study similarfigures, they learn that corresponding sides areproportional and corresponding angles are con-gruent. In this activity, they determine actual mea-surements of similar figures to explore these prop-erties. Students are asked to measure the anglesin the figures. They will need protractors to dothis. Alternatively, students could trace the figures,cut them out, and place one figure on top ofanother to compare angle measures.

Materials• Protractors

LESSON 3: SIMILAR POLYGONS

Hand out copies of the Similar Polygonsactivity sheet, p. 39. In this lesson, students aregiven pairs of similar figures to study. They usetheir knowledge of the properties of similar fig-ures to identify the corresponding sides andangles. Students also compare the measurementsof the sides to determine the scale factor for eachpair of similar figures. As an extension, you maywish to have students label the figures withappropriate unit measures.

LESSON 4: SPECIAL RIGHT TRIANGLES

Hand out copies of the Special RightTriangles activity sheet, p. 40. Provide centime-ter rulers to use with this activity. Through measur-ing and calculating ratios, students discover spe-cial characteristics of 30-60-90 and 45-45-90triangles. Students calculate the sine and cosinefor 30°, 60°, and 45° angles. Once studentshave completed the activity, ask them to comparethe sine of 60° to the cosine of 30° and thecosine of 60° to the sine of 30°. Discuss whythese values are equivalent. Talk about why theratios for the 45-45-90 triangle are equivalent.

LESSON 5: CONSTRUCTING SIMILARPOLYGONS

Hand out copies of the Constructing SimilarPolygons activity sheet, p. 41. In this lesson,students apply what they have learned about similarity to construct their own similar figures,given a scale factor. After constructing the figures,students may wish to label the sides with theappropriate measures.

1. If a person makes a donation of $35 to the charity mentioned in problem 2 above, how much money goes directly to the people the charity helps?________________________________

2. A 15-gram serving of crackers contains160 mg of sodium. How many milligrams of sodium are there per gram of crackers?________________________________

3. A car rental agency charges $10.75 to a customer for exceeding the daily mileage limit by 43 miles. What is the rate per mile for exceeding the daily mileage limit? ____________________________________________________


UNIT 4 LESSON 1

RATIO AND PROPORTION NAME __________________________

4. A convenience store sells an average of 72 plastic bottles of soda and 48 cans ____________________of soda daily.

5. A chef uses 6 pounds of shrimp and 4 pounds of scallops in a favorite recipe.________________________________

6. A package of fruit snacks contains 10 pouches and costs $1.99. __________________________________________

A. Directions: Write a ratio for each situation. Express the ratio in simplest terms.Round numbers when appropriate.

1. Socks are sold in packages of 3 pairs. The price of the package is $6.50.________________________________

2. A charity states that $0.75 of every dollar goes directly to the people thecharity helps. ____________________________________________________

3. A new development in town has 68 single-family homes and a complexwith 22 condominiums. ____________

B. Directions: Write proportions and solve the problems.

4. How many pounds of shrimp must becombined with 6.5 pounds of scallops to make the recipe mentioned in problem 5above? __________________________________________________________

5. A car travels 325 miles in 61/2 hours.What is the rate per hour? __________________________________________

6. Complete the chart to show the amountof each ingredient needed to make different serving sizes of the recipe.

6 Salmon Steaks

1 Clove of Garlic

1/3 Cup Butter

2 Onions

2 10 16

UNIT 4 LESSON 2

EXPLORING SIMILARITY NAME __________________________

Directions: Follow the instructions and answer the questions.

1. On the figures, write the lengths of the segments AB, AC, DE, and DF in units.

2. Use what you know about right triangles to find the lengths of BC and EF. Write themeasurements on the figures.

3. Write a ratio comparing each side of figure DEF to the corresponding side of figure figure ABC.

________________________________________________________________________

4. What can you say about the corresponding sides of the figures?__________________

________________________________________________________________________

5. Measure the angles of both figures. List all the pairs of congruent angles.

________________________________________________________________________

6. What can you say about the corresponding angles of the figures? ________________

________________________________________________________________________

7. Are the two figures similar? Why or why not? ________________________________

________________________________________________________________________

8. Draw a figure that is similar to figure ABC. Use a scale factor of 1/2.


B

A C

E

D F


UNIT 4 LESSON 3

SIMILAR POLYGONS NAME __________________________

Directions: Name the corresponding sides and angles for each pair of similar polygons. Then write the scale factor.

1. Corresponding sides:______________________________________________________

Corresponding angles: ____________________________________________________

Scale factor of ∆LMN to ∆OPQ: ________



Scale factor of figure LMNO to figure HIJK:____________________________________



Scale factor of ∆DEF to ∆ABC:__________

unit 4-2.eps

M

L N

P

QO

I J

KH

M N

L O

E F

D

6 cm6 cm

6 cmA

B C

4 cm

4 cm

4 cm


UNIT 4 LESSON 4

SPECIAL RIGHT TRIANGLES NAME __________________________


1. Measure the sides of triangles ABC and DEF. Write the measurements on the triangles.Are the figures similar? ____________________________________________________

2. Measure the sides of triangles DEF and JKL. Write the measurements on the triangles.Are the figures similar? ____________________________________________________

3. The sine (sin) of an angle equals the ratio of the opposite leg of that angle to thehypotenuse. In triangle ABC, AC is the leg opposite �B. Find the sine for all of the30° angles. ______________________________________________________________

4. Find the sine of all the 60° and 45° angles. ____________________________________

5. The cosine (cos) of an angle equals the ratio of the adjacent leg of that angle to the hypotenuse. In triangle ABC, AB is the adjacent leg to �B. Find the cosine for all the 30° angles. ______________________________________________________________

6. Find the cosine for all the 60° and 45° angles. __________________________________

30˚

60˚A

B

C

60˚

30˚

F D

E45˚

45˚

H

IG

45˚

45˚

KJ

L


UNIT 4 LESSON 5

CONSTRUCTING SIMILAR NAME __________________________POLYGONS

Directions: Use the given scale factor to draw a similar figure for each polygon.Label the similar figure using the same letters with prime signs (‘).

A

BC

H I

J

KL

M

unit 4-9.eps

O

P

Q

RN

DE

G

F

4. Scale factor: 1/2 2. Scale factor: 1/2

3. Scale factor: 31. Scale factor: 2



LESSON 1

A. Students may express ratios in different ways:as fractions, as a verbal expression, or separatedby a colon. Other ratios are possible. Sampleratios are provided.

1. $2.17 for 1 pair of socks

2. 3/4

3. Single-family homes to condominiums= 34 to 11

4. Bottles to cans = 3/2

5. Shrimp to scallops = 3/2

6. Cost per pouch � $0.20

B.1. 0.75/1 = x/35

x = $26.25

2. 15/160 = 1/x

15x = 160x = 10.7 mg

3. 10.75/43 = x/1

43x = 10.75x = $0.25

4. 3/2 = x/6.5

2x = 19.5x = 9.75 lb

5. 325/6.5 = x/1

6.5x = 325x = 50 mph

6.

LESSON 2

1 and 2.

3. DE = 6 = 2AB 3 1

DF = 8 = 2AC 4 1

EF = 10 = 2BC 5 1

4. The corresponding sides of figures ABC and DEF are proportional.

5. �A ��D, �B � �E, �C � �F

6. The corresponding angles of figures ABC andDEF are congruent.

7. Figures ABC and DEF are similar. The corresponding sides are proportional and the corresponding angles are congruent.

B

A C

E

D F

5

4

310

6

8

6 Salmon Steaks

1 Clove of Garlic

1/3 Cup Butter

2 Onions

2 10 16

23

1

313

13

2

323

19

59

89

3 5

1 2

LESSON 2, CONT.8.

LESSON 3

1. Corresponding sides: LM and OP, LN and OQ, MN and PQCorresponding angles: �L and �O,

�M and �P, �N and �QScale factor of ∆LMN to ∆OPQ: 1/2

2. Corresponding sides: HI and LM, IJ and MN,JK and NO, HK and LO

Corresponding angles: �H and �L, �I and �M, �J and �N, �K and �O

Scale factor of figure LMNO to figure HIJK: 3/1

3. Triangles ABC and DEF are equilateral, so all sides are proportional and all angles are congruent.

Scale factor of ∆DEF to ∆ABC: 11/2

LESSON 4

1. Measurements may vary. Triangles ABC andDEF are similar.

2. Measurements may vary. Triangles GHI andJKL are similar.

3. Measurements may vary. For the 30° angles,students should find a sine of about 0.5.

4. Measurements may vary. For the 60° angles,

students should find a sine of about 0.87. Forthe 45° angles, students should find a sine of about 0.71.

5. Measurements may vary. For the 30° angles, students should find a cosine of about 0.87.

6. Measurements may vary. For the 60° angles, students should find a cosine of about 0.5. For the 45° angles, students should find a cosine of about 0.71.

LESSON 5

1.

2.

3.

4.


UNIT – 4 ANSWER KEY

D' E'

F'

G'

H' I'

J'

K'L'

M'

O'

P'

Q'

R'N'

A'

B'

C'



UNIT 5 – CIRCLES

Lesson 1: Circle Construction

Lesson 2: More Circle Construction

Lesson 3: Parts of a Circle

Lesson 4: Label the Circle

Lesson 5: Applications with Circles


LESSON 1: CIRCLE CONSTRUCTION

Hand out copies of the Circle Constructionactivity sheet, p. 46. Often when students learnabout characteristics of figures, they read aboutthe figures and study illustrations. If students con-struct their own figures, they discover how thecharacteristics and properties actually work. Inthis lesson, students draw a circle, its diameter,and a chord. On the completed drawing, theyidentify a major and minor arc and a semicircle.As an extension, students can measure the partsof the circle they have drawn.

Materials• Compass for each student

LESSON 2: MORE CIRCLECONSTRUCTION

Hand out copies of the More CircleConstruction activity sheet, p. 47. This activityis a continuation of Lesson 1, and focuses oncentral angles, inscribed angles, tangents, andsecants. Students have likely read the definitionsof these circle elements many times. By actuallyconstructing the parts, the definitions will likelyhave more meaning to students.

Materials• Compass and protractor for each student

LESSON 3: PARTS OF A CIRCLE

Hand out copies of the Parts of a Circleactivity sheet, p. 48. Students use the circle on the sheet to reinforce recognition of the parts of

a circle and then to find different measurements(Lesson 4). You may wish to challenge studentsto list as many examples as they can for the different circle parts in the activity.

LESSON 4: LABEL THE CIRCLE

Hand out copies of the Label the Circleactivity sheet, p. 49. Have students take out theactivity sheet from Lesson 3. Explain that themeasures given in this lesson apply to the figurein Lesson 3 and that they will be labeling thisfigure. You may wish to review the properties that are used to find circle measurements. Whenstudents have completed the activity, invite volun-teers to present their methods for finding themissing measurements.

LESSON 5: APPLICATIONS WITHCIRCLES

Hand out copies of the Applications withCircles activity sheet, p. 50. In this activity, students use circle properties along with charac-teristics of other figures to find unknown measure-ments. Suggest that students first use the giveninformation to label the circle with as much detailas possible. Then students can study the labeledcircles to determine what methods they can use to find the missing measurements. Point out thatother figures such as triangles might be used intheir solutions.

UNIT 5 – CIRCLES


UNIT 5 LESSON 1

CIRCLE CONSTRUCTION NAME __________________________

Directions: Follow the instructions to construct a circle and parts of a circle.

1. Draw a circle with its center at X and a radius of 3 cm.

2. Draw a line connecting two points on the circle that passes through the center.Label the points A and B. What is this line called? ______________________________

3. What is the measure of AB in relationship to the whole circle? What is this arc called?__________________________________________________________________________

4. Label another point on the circle C. Name an arc that is less than half the circle. What is this arc called? ____________________________________________________

5. Name an arc that is more than half the circle. What is this arc called?__________________________________________________________________________

6. Draw a line that connects points A and C. Does this line pass through the center of thecircle? What is this line called? ______________________________________________

__________________________________________________________________________

•X

UNIT 5 LESSON 2

MORE CIRCLE CONSTRUCTION NAME __________________________

Directions: Follow the instructions to construct a circle and parts of a circle.

1. Draw a circle with a its center at Y and a radius of 1 inch.

2. Using Y as a vertex, draw and label an angle that measures 60°. Label points A and B where the sides of the angle intersect the circle. What is a name for this angle? Why?

__________________________________________________________________________

3. Given that the measure of �AYB is 60°, what is the measure of AB? What fraction of the circle is this? What kind of arc is AB? ______________________________________

__________________________________________________________________________

4. Label 2 points on the circle C and D. Draw an angle with C as a vertex and with sides AC and CD. What kind of angle is �ACD?

__________________________________________________________________________

5. Draw a line that intersects the circle only at point D. What is this line called?

__________________________________________________________________________

6. Draw two lines through the circle that intersect at a point on the exterior of the circle.Label the intersection point E and the lines FG and IH. What are these lines called?

__________________________________________________________________________


•Y


UNIT 5 LESSON 3

PARTS OF A CIRCLE NAME __________________________

Directions: Look at the circle below. Name one example of each of the following items:

1. Diameter _________

2. Radius _________

3. Central Angle _________

4. Inscribed Angle _________

5. Chord _________

6. Secant _________

7. Tangent _________

8. Semicircle _________

9. Minor Arc _________

10. Major Arc _________

ED

C

B

O

J

FG

H

I

A


UNIT 5 LESSON 4

LABEL THE CIRCLE NAME __________________________

Directions: Take out your activity sheet for Lesson 3. Use that sheet and the giveninformation on the left to find the missing measures on the right. Write your answerson the lines, and label the circle with the appropriate information.

Given Find

1. AD is a diameter Measure of �DFA __________

2. Measure of �DAF = 40° Measure of DF __________

Measure of �ADF __________

3. Measure of FG = 15° and measure Measure of AI __________of GI = 20°

4. Measure of �DJI = 97.5° Measure of AB __________

5. Measure of �CHB = 25° Measure of CB __________

Measure of CD __________

Measure of �CEF __________

1. Given:r = 5AC � FDOH � FDOG � ACAC = 8

Find: EH = __________

2. Given:AX = 6AC = 10DX = 9

Find: DB =

__________

3. Given:AD is tangent to O

at AOC = 12AD = 5

Find: OD =

________

4. Given:r = 7AC = 20EC = 15

Find: DE =__________

5. Given: BC = 9AB = 16

Find: DC =_________

6 Given:AB = 110°

Find: �ABC =________


UNIT 5 LESSON 5

APPLICATIONS WITH CIRCLES NAME __________________________

Directions: Solve for the unknown.

AB

C

D

x

A

GB

F

O E H

DC

A

B C

D

O

C

A

E

D

O

B

B

C

D

A

B C

A

X



LESSON 1

Placement of points and lines on students’ circlesmay vary. Sample is provided.

1.

2. See circle above. It is a diameter.3. AB is half of the circle. It is a semicircle.4. See circle above. BC is less than half the

circle. It is a minor arc.5. CAB is more than half the circle. It is a

major arc.6. See circle above. This line does not pass

through the center of the circle. It is a chord.

LESSON 2

Placement of points and lines on students’ circlesmay vary. Sample is provided.

1.

2. See circle. �AYB is a central angle since its vertex is at the center of the circle.

3. AB = 60°, which is 1/6 of the circle. AB is a minor arc.

4. See circle. �ACD is an inscribed angle.5. See circle. This line is a tangent to

the circle.6. See circle. FG and HI are secants.

LESSON 3

Students’ answers may vary. Samples are provided.1. DA2. OD3. �BOC4. �DFA or others5. DF or AF or AD6. CH or BH7. EC or EF8. DFA or DBA9. AB, BC, CD, DF, FG, GI, IA10. Some examples: DFB, FAC, IAC

LESSON 4

1. �DFA = 90°

2. DF = 80°

�ADF = 50°

3. AI = 65°

4. AB = 80°

5. CB = 70°

CD = 30°

�CEF = 70°

B

C

A

X

unit 5-9.eps

A

C

I

B

G

DE

FH

y60˚

4.

DE = 7

5. DC = 12

6.�ABC = 55°

C

A

E

D

O

B

6

7

7

(EC) (DC) = (AC) (BC) 15x = (20) (6)

x = 8 = DC DE = EC – DC

7 = 15 – 8



LESSON 5

1. EH = 2

2. DB = 11

3. OD = 13

A

GB

F

O E H

DC

4

4

4

4

5

3

AB

C

D

x

y6

48

3 = y6 4 = 8 y. B

C

D

A

16

9x

16 9 = x12 = x

2

.

B C

A

55˚

55˚

70˚

110

250

E

D

C

B

O

J

FG

I

A

unit 5-1answ.eps

70˚

80˚

40˚

65˚

97.5˚

50˚

80˚

90˚

20˚

25˚

H

15˚

70˚

30˚

LESSON 4, CONT.

A

B C

D

O

1212

5

5 + 12 = OD13 = OD

2 2 2


UNIT 6 – PERIMETER AND AREA

Lesson 1: Area of Parallelograms

Lesson 2: Area of Triangles

Lesson 3: Area of Trapezoids

Lesson 4: Area of Circles

Lesson 5: Recreation Area



LESSON 1: AREA OFPARALLELOGRAMS

Hand out copies of the Area ofParallelograms activity sheet, p. 56. Moststudents should be familiar with the formulas forthe area of a variety of polygons. Often theseformulas are memorized without really under-standing why they work. In this activity, studentscompare the areas of a rectangle and parallelo-gram that have the same base and height mea-surement. They will discover that by cutting a tri-angle off the end of the parallelogram, they canconstruct a rectangle. To further clarify the con-cept of area, you could have students examinethe number of squares inside the figures.

LESSON 2: AREA OF TRIANGLES

Hand out copies of the Area of Trianglesactivity sheet, p. 57. To many students, the formula for the area of a parallelogram mayseem obvious. It is easy to visualize multiplyingthe base by the height in a rectangle or square.The formula for the area of a triangle may not be quite as obvious. In this activity, through givenillustrations and students’ own constructions, theywill understand why the formula 1/2bh works.

LESSON 3: AREA OF TRAPEZOIDS

Hand out copies of the Area of Trapezoidsactivity sheet, p. 58. After students have complet-ed the lessons on parallelograms and triangles,they should be familiar with the concept ofmanipulating figures and using known formulasto find formulas for the areas of other figures. In

this lesson, students study two ways to test the formula for the area of a trapezoid. First,they compare the areas of a rectangle and atrapezoid that have the same base and height.Then students are asked to think of a way to test the formula for the area of a trapezoid using triangles.

LESSON 4: AREA OF CIRCLES

Hand out copies of the Area of Circles activitysheet, p. 59. Understanding the formula for thearea of a circle may present more difficultiesthan for polygons. This lesson presents a hands-on method for measuring circumference anddetermining the relationship between the diame-ter and circumference of a circle. Once this rela-tionship is established, students use a diagram ofsections of a circle to explore area. The diagramshows how the sections of a circle, all of whichhave sides equal to the radius, can be arrangedto form a parallelogram. Then, through substitu-tion, students find the formula for the area of acircle. You may find it useful to have studentsdivide a circle into equal sections, cut out the sections, and arrange them to look like the dia-gram on the activity sheet.

As students measure the circumference of the circles, you may want to suggest that they tapeone end of the string on the circle.

Materials• Centimeter rulers



LESSON 5: RECREATION AREA

Hand out copies of the Recreation Area activ-ity sheet, p. 60. In this lesson, students apply theformulas for the areas of the figures they haveexplored. The activity also includes opportunitiesfor measuring perimeter. Measurements are pro-vided, and in some cases students must combinethe areas of figures to find the area of a compos-ite figure. You may wish to have students createproblems of their own using the drawing on theactivity sheet.


UNIT 6 LESSON 1

AREA OF PARALLELOGRAMS NAME __________________________

Directions: Follow the instructions below and answer the questions.

1. Look at the rectangle above. How many units is the base? How many units is the height? __________________________________________________________________

2. What is the area of the rectangle?____________________________________________

3. Look at the parallelogram above. How is it different from the rectangle? ____________

__________________________________________________________________________

4. What is the base and height of the parallelogram? ______________________________

__________________________________________________________________________

5. Above the rectangle, draw a picture of the parallelogram with the shaded triangleremoved. Now imagine placing the shaded triangle on the right side of your drawing.Draw what this figure looks like.

6. How does the figure you drew compare with the rectangle? ______________________

__________________________________________________________________________

7. What is the area of the parallelogram?________________________________________

8. Using the grid above, try doing the same thing with another rectangle and parallelogram. The figures you use must have the same base and height. What can you conclude from this investigation?__________________________________________

y

x


UNIT 6 LESSON 2

AREA OF TRIANGLES NAME __________________________


1. Draw a diagonal of the rectangle.

2. Are the two triangles formed by the diagonals congruent? ________________________

3. What is the area of one of the triangles as compared to the area of the rectangle?

__________________________________________________________________________

4. Now draw a diagonal of the parallelogram.

5. What is the relationship of the two triangles formed by the diagonal? ______________

__________________________________________________________________________

6. What is the area of one of the triangles as compared to the area of the parallelogram?

__________________________________________________________________________

7. Draw some more examples of rectangles and parallelograms with diagonals.

8. What is the formula for the area of a rectangle or parallelogram? Based on thisformula and your investigation, what can you conclude about the formula for the areaof a triangle?______________________________________________________________

y

x


UNIT 6 LESSON 3

AREA OF TRAPEZOIDS NAME __________________________

Directions: Follow the instructions below and answer the questions.

1. Study the rectangle and trapezoid above. How are these figures similar?

________________________________________________________________________

2. How are the figures different? ______________________________________________

3. What is the area of the rectangle? __________________________________________

4. Using the formula A = 1/2 (b1 + b2)h, find the area of the trapezoid. _____________

5. Draw two triangles on the ends of the trapezoid so it has the same dimensions as therectangle. Find the area of each triangle. Compare the area of the new figure you have drawn with the area of the rectangle. Describe the results.

________________________________________________________________________

6. Draw another trapezoid on the grid above. Divide the trapezoid into two triangles.Find the area of each triangle. Explain how the formula for the area of a trapezoid can be derived from the formula for the area of a triangle. ______________________

________________________________________________________________________

________________________________________________________________________

y

x


UNIT 6 LESSON 4

AREA OF CIRCLES NAME __________________________


1. What is the diameter of circle A? Of circle B?

Diameter of A ____________ Diameter of B ____________

2. Using string or thread, find the approximate circumference of each circle.

Circumference of A� ____________ Circumference of B� ____________

3. Write a ratio of the circumference to the diameter for each circle. What do you find?

A._______________________________ B. _______________________________

________________________________________________________________________

4. Will this ratio be the same for all circles? Why or why not?______________________

________________________________________________________________________

5. Compare your ratios to the value of �. Based on your investigation, explain why theformula C = 2�r works. ___________________________________________________

________________________________________________________________________

6. The figure on the right above shows a circle divided into sections, with the sections cutout and placed side by side to form a parallelogram. The height represents the radiusof the circle. Since the sections are placed with opposite ends adjacent to one another,the base represents half the circumference. Use what you know about the area of aparallelogram to derive the formula for the area of a circle. (Hint: Substitute for thecircumference in the measure of the base.) ___________________________________

________________________________________________________________________

A

r = 2.5 cm

B

r = 2 cm

radi

us

1/2 Circumference


UNIT 6 LESSON 5

RECREATION AREA NAME __________________________

Directions: The drawing below shows plans for developing a park and recreationarea. Use the plans to answer the questions.

1. The playground will be fenced in. What is the total length of fencing needed for thisarea? ____________________________________________________________________

2. Developers need to know the area of the field to determine how much grass seed isneeded. What is this area? __________________________________________________

3. What is the total area of this park?_________________ If 1 acre is 43,560 square feet,approximately how many acres is the park?____________________________________

4. What is the total area set aside for forest and hiking trails, not including the duck pond?__________________________________________________________________________

5. To buy fertilizer for the gardens, developers need to know the area. What is the totalsquare feet in both gardens?_________________________________________________

6. The pool area will also be enclosed with a fence. What is the total length of fence needed?__________________________________________________________________

7. What is the area of the skating park? _________________________________________

8. Developers plan to plant grass in the picnic area, and they need to know the area ofcoverage. What is the area? ________________________________________________

9. What is the total area of the space surrounding the pool? _______________________

Duck Pondd = 150 ft

400 ft

Forest/Hiking Trails140 ft

460 ft

260 ft

150 ft

300 ft

Field

Garden

Pool

SkatingPark 140 ft

170 ft

PicnicArea

165 ft40 ft

Garden

15 ft

15 ft

25 ft15ft

85 ft

32 ft

50 ft

130

ftPlayground

130 ft

120 ft

170 ft 150 ft

600 ft

450 ft



LESSON 1

1. Base = 6 units; Height = 4 units2. 24 square units3. Answers may vary. Sample answer: The

parallelogram doesn’t have right angles. The parallelogram has a different shape than the rectangle.

4. Base = 6 units; height = 4 units5.

6. It has the same dimensions and the same shape as the rectangle.

7. 24 square units8. Students’ drawings may vary. Students should

conclude that the formula for the area of a parallelogram is the same as the formula for the area of a rectangle: A = bh.

LESSON 2

1. One of the two possible diagonals is shown.

2. Yes3. One of the triangles is half the area of the

rectangle.

4. One of the two possible diagonals is shown.

5. They are congruent triangles.6. One triangle is half the area of the

parallelogram.7. Students’ drawings will vary.8. The formula for the area of a rectangle or

parallelogram is A = bh. To find the area of atriangle, find half the area of a parallelogram formed by 2 congruent triangles. The formula is A = 1/2 bh.

LESSON 3

1. Sample answer: They have the same height. The base of the trapezoid is the same lengthas the base of the rectangle.

2. Sample answer: The rectangle has 2 pairs of parallel sides and the trapezoid has 1 pair of parallel sides. The rectangle has 4 right angles and the trapezoid has no right angles.

3. 7 units x 4 units = 28 square units4. 1/2 (7 + 3)4 = 20 square units



LESSON 3, CONT.

5.

The area of each triangle is1/2(2 x 4) = 4. So the area of the trapezoid + the area of both triangles = 20 + 4 + 4 or 28 squareunits. This is the area of the rectangle.

6. Students’ drawings and explanations may vary. Sample answer is provided.

The trapezoid is divided into 2 triangles, so the sum of the areas of the triangles equals the area of the trapezoid. Both triangles have the same height, so the formula makes sense.If you find the area of each triangle and add,you’ll get the same result as you would using the formula.

LESSON 4

Students’ answers and measurements may vary.Sample answers are provided.

1. Diameter of A = 5 cm; Diameter of B = 4 cm2. Circumference of A�15 cm;

circumference of B�12 cm3. A = 15/5; B = 12/4

4. As the radius of a circle changes, so does the circumference, so the ratio will be the same for all circles.

5. The value of ��3.14, which � the ratio of the circumference to the diameter. Since the circumference � 3 times the diameter and the radius = 1/2d, the circumference would be 2 • � • r.

6. The formula for the area of a parallelogramis base • height.

A = bhA = 1/2C • r (1/2C is the base of the

figure in the illustration)A = 1/22�r • r (Substitute 2�r for C)A = �r2 (Simplify)

LESSON 5

1. 570 ft2. 42,000 ft2

3. 225,000 ft2; a little over 5 acres4. 60,200 – 17,662.5 = 42,537.5 ft2

5. 2600 + 1440 = 4040 ft2

6. 410 ft7. 23,800 ft2

8. 21,450 ft2

9. 2875 + 1725 + 2(7500) = 6100 ft2

b2b1

hh


UNIT 7 – SOLIDS

Lesson 1: Characteristics of Solids

Lesson 2: Surface Area of Solids

Lesson 3: Volume

Lesson 4: Volume and Surface Area Applications


UNIT 7 – SOLIDS

LESSON 1: CHARACTERISTICS OFSOLIDS

Hand out copies of the Characteristics ofSolids activity sheet, p. 65, and the Netsactivity sheets, pp. 66 and 67. In this lesson, students cut out and fold nets to construct asquare pyramid, triangular prism, cylinder, andcone. Students use the solids to explore the vari-ous components and develop their own descrip-tions of each solid. These descriptions are basedon the characteristics of the figures.

Students need to save the solids to use inLessons 2 and 3. After students cut out thesolid nets, suggest that they fold the sides and not use tape. They may find it helpful in the otheractivities to fold and unfold the figures to examinedifferent aspects. Students might also look at thenumber of edges and vertices in the solids.

LESSON 2: SURFACE AREA OFSOLIDSHand out copies of the Surface Area ofSolids activity sheet, p. 68. Ask students to use the solids constructed for Lesson 1. Students will use the solids to calculate surfacearea. Students can manipulate the figures andturn them to see exactly what each face looks like and to find the dimensions in units. Besideshelping to develop understanding of surfacearea, working with manipulatives will help stu-dents visualize the parts of solids usually shownin illustrations. Some students may need to reviewconcepts relating to areas of polygons. Make sure students understand that surface area ismeasured in square units.

LESSON 3: VOLUME

Hand out copies of the Volume activity sheet, p. 69. Have students use the solids constructed for Lesson 1. At the beginning of this lesson,students identify the formula for finding the vol-ume of a rectangular prism. Most students willunderstand why the formula makes sense for afigure shaped like a box. Volume of other solidscan be related to the volume of a rectangularprism. Since other prisms and cylinders have parallel, congruent bases, multiplying the area of the base by the height will give the volume ofall prisms and cylinders. Remind students that volume is measured in cubic units.

Problems 3 and 4 state that the volume of asquare pyramid and cone are equal to 1/2 the volume of a cube and cylinder (respectively) withcongruent bases. Students might enjoy provingthis fact with solids constructed from heavy-stockpaper and material such as sand, sugar, or rice.Students can fill the square pyramid and conewith the material, then pour it from the pyramidinto the cube and from the cone into the cylinder.Students can count the number of times they mustdo this to fill the cube and cylinder.

LESSON 4: VOLUME AND SURFACEAREA APPLICATIONS

Hand out copies of the Volume and SurfaceArea Applications activity sheet, p. 70. In thislesson, students apply the formulas they havelearned for finding the surface area and volumeof prisms, pyramids, cylinders, and cones. Toextend this activity, find objects in the classroomthat students can measure to determine surfacearea or volume.

UNIT 7 LESSON 1

CHARACTERISTICS OF SOLIDS NAME __________________________

Directions: Cut out the nets from the Nets activity sheets. Fold the nets to form solidfigures. Make sure the letter labels are on the outside of the figures when you fold.Use the figures to follow the instructions and answer the questions.

1. Study solid figures A and B and complete the chart.

2. This solid is a triangular pyramid.

Using this illustration, the information above, and your own knowledge of solids, what are the characteristics of pyramids? __________________________________________

__________________________________________________________________________

3. This solid is a pentagonal prism.

Using this illustration, the information above, and your own knowledge of solids, whatare the characteristics of prisms? _____________________________________________

__________________________________________________________________________

4. Study figures C and D. How are they similar? How do they differ? _________________

__________________________________________________________________________

5. What are the characteristics of cones? ________________________________________

__________________________________________________________________________

6. What are the characteristics of cylinders?______________________________________

__________________________________________________________________________


Figure Name Bases: Number and Description

Faces: Number andDescription

A

B


UNIT 7 LESSON 1

NETS – SQUARE PYRAMID AND NAME __________________________TRIANGULAR PRISM

A

B

Directions: Cut along the solid lines. Fold on the dotted lines.


UNIT 7 LESSON 1

NETS - CYLINDER AND CONE NAME __________________________

C

D

Directions: Cut out shapes C and D along the solid lines, leaving the “circles” attached.Fold C into cone shape, and D into cylinder shape.


UNIT 7 LESSON 2

SURFACE AREA OF SOLIDS NAME __________________________

Directions: Use the figures you constructed in Lesson 1. Follow the instructions andanswer the questions.

To find surface area, you can look at the constructed solid, or you can flatten out the solidand look at the figures that make up the faces and bases.

1. Look at the net for the cylinder. What figures make up this solid? Find the surfacearea and show your reasoning.

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

2. Look at the triangular prism you constructed. Describe how you would find the surfacearea of this figure. Calculate the surface area.

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________


UNIT 7 LESSON 3

VOLUME NAME __________________________

Directions: Use the figures you constructed in lesson 1. Follow the directions andanswer the questions.

1. How do you find the volume of a rectangular prism? ____________________________

________________________________________________________________________

2. What are the bases of the triangular prism you constructed? How would you findthe volume of this figure? What is the volume?

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

3. Look at the square pyramid you constructed. Imagine that you had a cube with a base the same size as the base of this pyramid. If you filled each solid, you would find that the pyramid has a volume that is 1/3 the volume of the cube. What formula can you use to find the volume of a pyramid? Find the volume of the square pyramid.

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

4. Look at the cylinder and cone you constructed. These figures have the same base and height. If you filled each solid, you would find that the volume of the cone is 1/3

the volume of the cylinder. What formulas can you use to find the volume of a cylinderand a cone? Find the volume of your cone and cylinder.

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________


UNIT 7 LESSON 4

VOLUME AND SURFACE NAME __________________________AREA APPLICATIONS

A. Directions: Find the surface area. Use 3.14 for �.1. 3.

2. 4.

B. Directions: Solve the problems. Round to the nearest hundredth when appropriate.

1. What is the volume of the cube in number 1 above?____________________________

________________________________________________________________________

2. Find the volume of a cone whose base has a radius of 4 inches and a height of 51/2 inches.

________________________________________________________________________

3. What is the volume of the triangular prism in number 4 above? __________________

________________________________________________________________________

4. What is the volume of a soup can whose base has a diameter of 6.6 cm and a height of 10 cm? _______________________________________________________________

________________________________________________________________________

5. A cereal box has a length of 71/2 inches, a width of 21/4 inches, and a height of 103/4

inches. What is the volume of the cereal box? _________________________________

________________________________________________________________________

6. What is the volume of the cylinder in Part A, number 3 above? _________________

________________________________________________________________________

3 cm3 cm

3 cm

10 ft4 ft

4 ft

16 m

8 m

16 m

40 m6 m

10 m

Surface Area =

__________

Surface Area =

__________

Surface Area =

__________

Surface Area =

__________



2. The faces of pyramids are triangles. Thefaces meet at a point. The shape of the basegives the solid its name.

3. There are 2 bases that are congruent andparallel. The bases are joined by rectangularfaces. The shape of the bases gives the solid its name.

4. Both figures have bases that are circles. The cylinder has 2 bases, and the cone has 1 base. Both solids have curved surfaces.

5. A cone has a circular base, and the surface meets at a point somewhere above or below the base.

6. A cylinder has 2 circular bases that are congruent. The bases are joined by a curved surface, which when flattened forms arectangle.

LESSON 2

1. A rectangle and 2 circles. Add the area of the rectangle to 2 times the area of one of the circles.

2. Find the area of one of the bases (A =1/2 bh). Double this figure and add it to the areas of the 3 faces.

LESSON 3

1. Multiply the area of the base by the height.2. Triangles. Find the area of the triangular

base and multiply by the height.3. V = 1/3Bh.4. Volume of a cylinder = Bh.

Volume of a cone = 1/3Bh.

LESSON 4

A. 1. 54 cm2

2. 192 ft2

3. 1205.76 m2

4. 1536 m2

B. 1. 27 cm3

2. 92.11 in.3

3. 1920 m3

4. 341.95 cm3

5. 181.41 in.3

6. 3215.36 m3

Figure Name Bases: Number and Description

Faces: Number andDescription

A square pyramid

B triangular prism

1 base; square

2 bases; congruent triangles

4 faces; all triangles

3 faces; all rectangles

LESSON 11.


UNIT 8 – COORDINATE GEOMETRY

Lesson 1: Distance Formula

Lesson 2: Midpoint Formula

Lesson 3: Quadrilaterals on Coordinate Grids

Lesson 4: More Quadrilaterals on Coordinate Grids

Lesson 5: Equation of a Line


UNIT 8 – COORDINATE GEOMETRY

LESSON 1: DISTANCE FORMULAHand out copies of the Distance Formulaactivity sheet, p. 74. In this activity, students test the distance formula to find lengths of differ-ent line segments. First, students count the inter-vals to find the lengths of line segments that areparallel to the horizontal and vertical axes. Afterthey have determined the length by using thismethod, they use the distance formula to find thesame length. Through this investigation, studentsdiscover that the distance formula is useful forfinding lengths of segments that are not parallelto either axis.

LESSON 2: MIDPOINT FORMULAHand out copies of the Midpoint Formulaactivity sheet, p. 75. Students will likely use themidpoint formula often in problem-solving situa-tions. This activity shows on a coordinate planewhy the formula works. Students use the slopeand distance formula to prove that the line con-necting the midpoints of two sides of a triangle isparallel to the third side and equal to half itslength. You may wish to have students test the for-mula with a triangle that is not a right triangle.They may do this with or without a coordinategrid. Suggest that students use measurements thatwill be easy to work with, such as even numbers.

LESSON 3: QUADRILATERALS ONCOORDINATE GRIDSHand out copies of the Quadrilaterals onCoordinate Grids activity sheet, p. 76. Thisactivity reviews several important concepts.

Students first practice plotting ordered pairs. They then identify the figure visually and name its characteristics. Using these characteristics, students devise a plan to prove that the figure onthe grid has all of the necessary characteristics.Finally, students apply the slope formula intro-duced in this unit to determine parallel sides ofthe figure.

LESSON 4: MORE QUADRILATERALSON COORDINATE GRIDSHand out copies of the More Quadrilateralson Coordinate Grids activity sheet, p. 77.This lesson is similar to the activity in Lesson 3.Students apply formulas to prove that the figurethey plot is a parallelogram. This process involvesusing the distance formula to find congruent sidesand the slope formula to find parallel sides.Students also use the distance formula to provethat the diagonals of the figure bisect each other.As an extension, students can draw other quadri-laterals and use the characteristics to prove theclassification of the figure.

LESSON 5: EQUATION OF A LINEHand out copies of the Equation of a Lineactivity sheet, p. 78. Students create tables ofordered pairs, then use them to graph the equa-tions of lines. Students calculate the slope of eachline and then study the graph to define slope intheir own words. As an extension, you may wishto have students investigate graphing differentkinds of equations as well as using the graph tosolve two linear equations.


UNIT 8 LESSON 1

DISTANCE FORMULA NAME __________________________

Directions: Follow the instructions andanswer the questions.

1. Plot the points A(1,7) and B(7,7).

2. In units, what is the length of AB?

________________________________

3. Using the distance formula

�(x2 – x1)2 + (y2 – y1)2 , find the

length of AB. _____________________

________________________________

________________________________

4. Plot the points C(4,0) and D(4,4).

5. In units, what is the length of CD? _____________________________________________

6. Use the distance formula to find the length of CD.________________________________

__________________________________________________________________________

7. Based on this investigation, when would the distance formula be most useful? ________

__________________________________________________________________________

8. Draw lines to form segments AD and BD.

9. What type of triangle does figure ABD appear to be? ____________________________

______________________________________________________________________________

10. Use the distance formula to find the measures of AD and BD. Describe the results.

__________________________________________________________________________

y

x


UNIT 8 LESSON 2

MIDPOINT FORMULA NAME __________________________


1. Plot the points A(0,0), B(0,8), andC(6,0).

2. Draw lines to form segments AB, BC,and AC. What kind of figure is this?

_________________________________

3. Find the midpoint of AB. Label thepoint D. What are the coordinates?

_________________________________

4. Find the midpoint of AC. Label the point E. What are the coordinates?

_________________________________

5. Draw a line to form segment DE.

6. Calculate the slope and length of DE. _________________________________________

________________________________________________________________________

7. Calculate the slope and length of BC. _________________________________________

________________________________________________________________________

8. Compare the slope and length of DE and BC. __________________________________

9. Describe the segment that connects the midpoints of two sides of a triangle.

________________________________________________________________________

________________________________________________________________________

10.Construct a different triangle. Select convenient coordinates. Are the results the same?

________________________________________________________________________

y

x


UNIT 8 LESSON 3

QUADRILATERALS ON NAME __________________________COORDINATE GRIDS


1. Plot the points A(2,2), B(4,6), C(9,6),and D(12, 2).

2. Draw lines to form segments AB, BC,CD, and AD.

3. What kind of figure does this appear to be?________________________________

4. What are the characteristics of the figureyou named in question 3?

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

5. Describe how you would prove what kind of figure ABCD is.

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

6. Follow the procedure you described in number 5. Explain your results.

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

y

x


UNIT 8 LESSON 4

MORE QUADRILATERALS NAME __________________________ON COORDINATE GRIDS


1. Plot the points A(2,3), B(4,11),C(12,11), and D(10,3).

2. Draw lines to form segments AB, BC,CD, and AD.

3. Prove that this figure is a parallelogram. What do you know about the lengths of the sides of a parallelogram?____________________

________________________________

________________________________

4. Find the lengths of the sides of figure ABCD. Describe the results.__________________

________________________________________________________________________

5. What else do you know about the sides of a parallelogram?______________________

________________________________________________________________________

6. Find the slope of the sides of figure ABCD. Describe your results. __________________

________________________________________________________________________

7. What do you know about the diagonals of a parallelogram? _____________________

________________________________________________________________________

8. Draw the diagonals for figure ABCD. Label the point of intersection E. Find the lengthof segments BE, ED, AE, and EC. Describe the results. ___________________________

________________________________________________________________________

9. Do you think that figure ABCD is a parallelogram? Explain. ______________________

________________________________________________________________________

y

x


UNIT 8 LESSON 5

EQUATION OF A LINE NAME __________________________


y

1. Graph the line of the equation y = 3x + 1.a. Substitute values of x to find y. Complete the table of ordered pairs.

b. Plot the ordered pairs. Connect the points with a line. Label the line R. Use two points to calculate the slope.________________

________________________________

________________________________

2. Graph the line of the equation y = 2x – 1. a. Substitute values of x to find y. Complete the table of ordered pairs.

b. Plot the ordered pairs. Connect the points with a line. Label the line S. Use two points tocalculate the slope.__________________________________________________________

________________________________________________________________________

3. Follow the same steps to graph the line of the equation y = x + 2. Label the line T. Find the slope.

________________________________

________________________________________________________________________

4. Study the graphs of the lines and the slopes. How would you describe “slope’’?

__________________________________________________________________________________

________________________________________________________________________

y

x

1 2 3 4

xy

1 2 3 4

xy

1 2 3 4

x



LESSON 11. See coordinate grid above.2. 6 units3. d = � (7 – 1)2 + (7 – 7)2

d = 64. See coordinate grid above.5. 4 units6. d = � (4 – 4)2 + (4 – 0)2

d = 47. The distance formula would be useful to find

the length of a segment that is not parallel toeither axis.

8. See coordinate grid above.9. ∆ABD appears to be an isosceles triangle.10. BD = � (7 – 4)2 +(7 – 4)2

BD = �18 = 3�2

AD = � (1 – 4)2 +(7 – 4)2

AD = �18 = 3�2

Since AD = BD, ABD is an isosceles triangle.

LESSON 21. See coordinate grid above.2. See coordinate grid above. ABC is a right

triangle.3. See coordinate grid above. D=(0,4)4. See coordinate grid above. E=(3,0)5. See coordinate grid above.

6. Slope of DE = 4 – 0 = – 40 – 3 3

Length of DE: � (0 – 3)2 + (4 –0)2

DE = 5

7. Slope of BC = 8 – 0 = – 40 – 6 3

Length of BC: � (0 – 6)2 +(8 –0)2

BC = 108. The slopes of DE and BC are equal. The

length of DE is half the length of BC.9. The segment that connects the midpoints of

two sides of a triangle is parallel to the third side and is half the length of the third side.

10. Drawings will vary. See sample triangle incoordinate grid above. Students should find the same results as in triangle ABC.

y

x

A (1,7) B (7,7)

C (4,0)

D (4,4)

y

x

B (0,8))

D (0,4)

E (3,0) C (6,0)

H (9,9)

J (9,7)

G (9,5)

(7,7) I

(5,5) F

Slope IJ = 0Length IJ = 2

Slope FG = 0Length FG = 4

A (0,0)

LESSON 4

1. See coordinate grid above.2. See coordinate grid above.3. A parallelogram has two pairs of

congruent sides.4. AB = � (4 – 2)2 + (11 – 3)2

AB = �68

BC = 8

CD = � (12 – 10)2 + (11 – 3)2

CD = �68

AD = 8Figure ABCD has two pairs of congruentsides.

5. A parallelogram has two pairs of parallelsides.

6. Slope AB = 11 – 3 = 8 = 44 – 2 2 1

Slope BC = 11 – 11 = 012 – 4

Slope CD = 11 – 3 = 8 = 4 12 – 10 2 1

Slope AD = 3 – 3 = 010 – 2



LESSON 3

1. See coordinate grid above.2. See coordinate grid above.3. This figure appears to be a trapezoid.4. A trapezoid has 4 sides and has one pair

of parallel sides.5. You can count the number of sides in figure

ABCD to determine it has 4 sides, then usethe slope formula to find out if any two sides are parallel.

Slope of AB = 6 – 2 = 24 – 2 1

Slope of BC = 6 – 6 = 09 – 4

Slope of CD = 6 – 2 = –4

9 – 12 3

Slope of AD = 2 – 2 = 012 – 2

The slopes of AB and AD are equal, so these segments are parallel. Figure ABCD is a trapezoid.

y

x

B (4,6)

A (2,2)

C (9,6)

D (12,2)

y

x

B (4,11)

A (2,3) D (10,3)

C (12,11)

E (7,7)

6.



LESSON 4, CONT.

The slopes of AB and CD are equal, so these sides are parallel. The slopes of BC and AD are equal, so these sides are parallel.

7. The diagonals of a parallelogram bisect oneanother.

8. See the coordinate grid on previous page.

BE = � (4 –7)2 +(11 – 7)2

BE = 5

DE = � (10 – 7)2 + (3 – 7)2

DE = 5

AD = � (7 – 2)2 +(7 – 3)2

AD = �41

CE = � (2 – 7)2 + (11 – 7)2

CE = �41

BE � DE and AE � CE

9. Figure ABCD is a parallelogram. Figure ABCD has two pairs of congruent sides andtwo pairs of parallel sides. The diagonals of figure ABCD bisect each other.

LESSON 5

1a.

1b. See coordinate grid above. Slope = 31

2a.

2b. See the coordinate grid above. Slope = 21

3.

Slope = 1

14. Answers may vary. Sample answer:

Slope is a ratio that compares the vertical change in the line to the horizontal change.

y

x1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

T

S

R

x

x

x

y

y

y

1 2 3 44 7 10 13

11

23

3 45 7

1 2 3 43 4 5 6


UNIT 9 – TRANSFORMATIONAL

GEOMETRY

Lesson 1: Translations and Rotations

Lesson 2: Reflections

Lesson 3: Transformations

Lesson 4: Translations and Reflections on a Coordinate Plane

Lesson 5: Rotations on a Coordinate Plane


UNIT 9 – TRANSFORMATIONAL GEOMETRY

LESSON 1: TRANSLATIONS ANDROTATIONS

Hand out copies of the Translations andRotations activity sheet, p. 84. In this activity,students explore the properties of translations and rotations. An example of a translation and a rotation are shown on dot paper. These exam-ples provide clear visual examples of the move-ments that have taken place in each transforma-tion. The illustrations also make it easy to see corresponding parts of each figure and its image.Students also construct their own translations and rotations.

Materials• Protractor to measure some angles

LESSON 2: REFLECTIONS

Hand out copies of the Reflections activitysheet, p. 85. As in Lesson 1, students examineand measure congruent figures to discover prop-erties of transformations. This activity presentsboth horizontal and vertical reflections. Studentsalso construct a reflection. After the constructionis completed, encourage students to study all fourfigures on the activity sheet and determine whichfigures are rotational images of one another.

LESSON 3: TRANSFORMATIONS

Hand out copies of the Transformationsactivity sheet, p. 86. Students apply what theyknow about different kinds of transformations tocompare figures and identify how they relate toone another. One figure in the activity relates to

the other figures not a as transformation, but as a similar figure. Students find similar figures anddetermine the scale factor. Once students havecompleted this lesson, it may be interesting tostudy all the figures and look for relationships.

LESSON 4: TRANSLATIONS ANDREFLECTIONS ON A COORDINATEPLANE

Hand out copies of the Translations andReflections on a Coordinate Plane activitysheet, p. 87. In previous lessons, students workedwith transformations on dot paper. This activitycombines concepts in transformational and coor-dinate geometry. Students see that constructingtranslations and reflections on a coordinate planeis very similar to doing transformational draw-ings on dot paper. Once the constructions havebeen completed, students list the coordinates forpoints on all figures. Take advantage of thisopportunity to relate the visual patterns createdby the transformations to the number patterns inthe coordinates. Encourage students to comparethe coordinates to look for these number patterns.

LESSON 5: ROTATIONS ON ACOORDINATE PLANE

Hand out copies of the Rotations on aCoordinate Plane activity sheet, p. 88. Againin this activity, students construct transformationson a coordinate plane. Students use the origin forthe turn center. Have students study the coordi-nates to find patterns among the original figureand the rotations.


UNIT 9 LESSON 1

TRANSLATIONS AND ROTATIONS NAME __________________________


1. Fill in the blanks: AB � _____ BC � _____ CD � _____ AD � _____

2. What is the relationship between the anglesin figures ABCD and A’B’C’D’? __________

3. How would you describe the translation of figure ABCD to A’B’C’D’? ______________

________________________________________________

4. Describe the rotation illustrated by figure EFG and image E’F’G’, using both direction and degrees. ______________________________________________________________

5. Are corresponding segments in a 180° rotation and its image parallel? __________________________________

6. Fill in the blanks:�FGE �______�GEF �______�EFG �_____

7. Translate figure QRST 2units down and 5 unitsleft. Draw the figure andmark congruent anglesand segments.

8. Draw a 90° counterclockwise rotation of figure HIJKL around point P. Label the rotationimage H’I’J’K’L’.

9. Using a dotted line, draw the following segment pairs: IP and I’P, HP and H’P, JP and J’P.

10.Measure these angles: �IPI’, �HPH’, �JPJ’. How do the angle measures relate to thedegrees in the rotation? ____________________________________________________

A

BC

D

A'

B'C'

D'

unit 9-2.eps

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unit 9-3.eps

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UNIT 9 LESSON 2

REFLECTIONS NAME __________________________


1. Which figures above show a figure and its horizontal reflection image? Which figuresshow a figure and its vertical reflection image? _________________________________

__________________________________________________________________________

2. How do segments connecting corresponding points on a figure and its image relate tothe reflection line?

__________________________________________________________________________

3. Using dotted lines, draw segments from each point of ABD and A’B’D’ to the vertical reflection line.

4. What do you find about the distance between the reflection line and a point of a figureand the distance between the reflection line and the corresponding point of its image?

__________________________________________________________________________

5. Which lines are parallel in a figure and its reflection image?______________________

__________________________________________________________________________

6. Draw a horizontal reflection image for A’B’C’D’E’. Label the image A”’B”’C”’D”’E”’.

7. How are corresponding points on this figure and the reflection image related?

__________________________________________________________________________

8. Describe the relationship between figures A’B’C’D’E’ and A”B”C”D”E”. ____________

__________________________________________________________________________86© Vivendi Universal Publishing and/or Math Blaster® Geometry

its subsidiaries. All Rights Reserved.

D

BC

E A

B' C'

D'

E'A'

E"

D"

A"

B"C"

•


UNIT 9 LESSON 3

TRANSFORMATIONS NAME __________________________

Directions: Answer thequestions.

1. How are figures QPRST andQ’P’R’S’T’ related?

________________________

________________________

________________________

2. Figure UVWX is similar to what other figures?__________________________________

3. Figures IJKL and I’J’K’L show what kind of a transformation? ______________________

__________________________________________________________________________

4. How are figures MNOPQ and TSRPQ related?__________________________________

__________________________________________________________________________

5. What is another name for segment QP? ________________________________________

6. How are figures Q’P’R’S’T’ and Q”P”R”S”T” related? ____________________________

7. What is one example of a horizontal reflection and one example of a vertical reflection?________________________________________________________________

__________________________________________________________________________

8. What is the scale factor of similar figures BCDA and UVWX? ____________________

9. What is the relationship between figures ABCD and LIJK? ________________________

__________________________________________________________________________

10. List the corresponding segments in all figures congruent to ABCD. _________________

__________________________________________________________________________

__________________________________________________________________________

X W

VUC' B'

D' A'

M N

OQ P

R

ST

S" T"

Q"P"

R"

B

A D

C

I J

LK'

J' I'

Q'

T' S'

R'

P'K

UNIT 9 LESSON 4

TRANSLATIONS AND NAME __________________________REFLECTIONS ON ACOORDINATE PLANE

Directions: Answer the questions.

1. Plot and label the following points:A(1,0); B(6,3); C(7,8); D(3,10);E(0,5).

2. Draw AB, BC, CD, DE, and ______EA. What kind of a figure is this? ________

________________________________

3. Draw an x-axis reflection of figure ABCDE.

4. Draw a y-axis reflection of figure ABCDE.

5. Draw a vertical translation for any figure.

6. Draw a horizontal translation for any figure.

7. Draw a horizontal/vertical translation for any figure.

8. Complete the table:

Transformation Coordinates of Pointsoriginal A(1,0); B(6,3); C(7,8); D(3,10); E(0,5)

x-axis reflection

y-axis reflection

vertical translation

horizontal translation

horizontal/vertical translation


UNIT 9 LESSON 5

ROTATIONS ON A COORDINATE NAME __________________________PLANE

Directions: Follow the instructions.

1. Plot the following points: P(1,1); Q(7,1); R(1,5).

2. Draw lines to form PQ, PR, and QR.

3. What kind of figure is this?__________________________________________________

4. Draw a 90° rotation of figure PQR around the origin. Label the figure P’Q’R’.

5. Draw a 180° rotation of figure PQR around the origin. Label the figure P”Q”R”.

6. Draw a 270° rotation of figure PQR around the origin. Label the figure P’’’Q’’’R’’’.

7. Complete the table.

Transformation Coordinates of Points

original P(1,1); Q(7,1); R(1,5)

90° rotation

180° rotation

270° rotation



LESSON 1

1. AB � A’B’, BC � B’C’, CD � C’D’, AD � A’D’2. Corresponding angles in the two figures are

congruent.3. Figure ABCD is translated 6 units to the right

and 1 unit up.4. E’F’G’ is a 180° clockwise rotation of

figure EFG around point P.5. Yes.6. �FGE��F’G’E’, �GEF��G’E’F’,

�EFG��E’F’G’

7. See figure Q’R’S’T’ above.8. See figure I’J’K’L’ above.9. See figures IJKL and I’J’K’L’ above.10. The angles measure 90°, which is the number

of degrees in the rotation.

LESSON 2

1. Figures ABCD and A”B”C”D”E” show a figure and its horizontal reflection image. Figures ABCDE and A’B’C’D’E’ show a figure and its vertical reflection image.

2. The segments are perpendicular to the reflection line.

3. See the diagram in the next column.4. The distances between the reflection line and

corresponding points of a figure and itsreflection image are equal.

5. The parallel lines in a figure and its reflectionimage are those lines that are parallel to thereflection line.

6. See A”’B”’C”’D”’E”’ in the diagram below.7. Corresponding points on the figures are the

same distance from the reflection line.8. A”B”C”D”E” is a 180° rotation of figure

A’B’C’D’E’.

LESSON 3“

1. Figure Q’P’R’S’T’ is a horizontal/vertical translation of QPRST.

2. Figure UVWX is similar to figures ABCD,A’B’C’D’, IJKL and I’J’K’L’.

3. A 180° rotation.4. MNOPQ is a horizontal reflection image

QPRST.5. QP is a reflection line.6. Q”P”R”S”T”is a 180° rotation of Q’P’R’S’T’.7. Figures ABCD and A’B’C’D’ show a vertical

reflection. Figures MNOPQ and QPRST showa horizontal reflection.

8. The scale factor of UVWX to BCDA is 2 to 1.9. ABCD and LIJK are translation images.10. Corresponding segments to AB are A’B’, IL,

and I’L’. Corresponding segments to BC are B’C’, IJ, and I’J’. Corresponding segments to CD are C’D’, JK and J’K’. Corresponding segments to AD are A’D’, LK, and L’K’.


unit 9-7.eps

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KJ

I

J'

K' L'

H'

I'P

R'

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D'

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Lesson 2, Problems 3 and 6

Transformation Coordinates of Pointsoriginal A(1,0); B(6,3); C(7,8); D(3,10); E(0,5)x-axis reflectionfigure AB’C’D’E’ A(1,0); B’(6,-3); C’(7,-8); D’(3,-10); E’(0,-5)y-axis reflectionfigure A”B”C”D”E A”(-1,0); B”(-6,3); C”(-7,8); D”(-3,10); E(0,5)vertical translationfigure KLMNO K(-1,2); L(-6,5); M(-7,10); N(-3,12); O(0,7) horizontal translationfigure FGHIJ F(4,0); G(9,-3); H(10,-8); I(6,-10); J(3,-5)horizontal/vertical translationfigure PQRST P(-4,-2); Q(1,-5); R(2,-10); S(-2,-12); T(-5,-7)



Answers may vary for problems 5-7.Sample answers are provided. For coordinate points, see below.

1. See figure ABCDE.2. See figure ABCDE. This figure is a

pentagon.3. See figure AB’C’D’E’.4. See figure A”B”C”D”E.5. See figure KLMNO.6. See figure FGHIJ.7. See figure PQRST.

P

T

S

R D' I

H

G

C'

B'

JQE'

AF

A"

B

C

D

E

O

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D"M

C"

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L

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LESSON 4

8.

Transformation Coordinates of Points

original P(1,1); Q(7,1); R(1,5)

90° rotationfigure P’Q’R’ P’(-1,1); Q’(-1,7); R’(-5,1)180° rotationfigure P”Q”R” P’’(-1,-1); Q’’(-7,-1); R’’(-1,-5)270° rotationfigure P”’Q”’R”’ P”’(1,-1); Q”’(1,-7); R”’(5,-1)



LESSON 5

The positions of rotation images students draw for problems 4-6 mayvary. Sample answers are provided.

1. See figure PQR.2. See figure PQR.3. A right triangle.4. See figure P’Q’R’.5. See figure P”Q”R”.6. See figure P’’’Q’’’R’’’.

R

QPP'''

Q'''

R'''P"

R"

Q"

P'R'

Q'

7.


UNIT 10 – REASONING AND PROOF

Lesson 1: Logical Reasoning

Lesson 2: Cut-and-Paste Proofs


LESSON 1: LOGICAL REASONING

Hand out the Logical Reasoning activitysheet, p. 94. This is a geometry game of drawingconclusions. It is played by revealing only oneclue at a time. Students should use a separatesheet of paper to cover subsequent clues. Afterreading each clue, students discuss possible solu-tions with a partner. When students determine thegeometric figure that is being described, theround ends.

After playing several rounds, students can createtheir own puzzles.

LESSONS 2A, 2B, AND 2C: CUT-AND-PASTE PROOFS

Hand out copies of the Cut-and-Paste Proofsactivity sheets, pp. 95–97. Students work withpartners to construct proofs using the cutouts pro-vided. Each statement and reason should be aseparate cutout. Students may have difficultysequencing a proof; there is generally not onlyone way to sequence a proof.

UNIT 10 – REASONING AND PROOF


ACTIVITY 1: WHAT AM I?

1. I am a quadrilateral.2. I have two congruent sides.3. I have two parallel sides.4. My parallel sides are not congruent.5. My base angles are congruent.6. I am convex.

I am a: _______________Solved at step: _________


1. I am a quadrilateral.2. My diagonals are congruent.3. My angles are right angles.4. My diagonals are perpendicular.5. My opposite sides are congruent.6. I am a rhombus.7. I am a rectangle.



1. I am a polygon.2. I am convex.3. My sides are all congruent.4. My angles are all congruent.5. My exterior angles measure 60°.6. My interior angles measure 120°.7. My diagonals are all congruent.


ACTIVITY 4: WHAT AM I?1. I have two congruent angles.2. My congruent angles each measure 45°.3. I have three sides.4. I have two congruent sides.5. My vertex angle measures 90°.6. My congruent sides measure 5” each.


ACTIVITY 5: CREATE YOUR OWNRIDDLE. WHAT AM I?

1. ________________________________

2. ________________________________

3. ________________________________

4. ________________________________

5. ________________________________

6. ________________________________

7. ________________________________


UNIT 10 LESSON 1

LOGICAL REASONING NAME __________________________

Directions: Reveal only one clue at a time by covering the others with a piece ofpaper. With a partner, try to determine which geometric figure is being described.

Given: �PAD ��QBC, AE �BEProve: DA �CB

Statements Reasons

�PAD ��QBC, AE �BE Given

AB �AB Reflexive Property

�PAD and �DAB form a straight �. Definition of straight angle�QBC and �CBA form a straight �.

�PAD and �DAB are supplementary. If two �s form a straight � , they are supple-�QBC and �CBA are supplementary. mentary.

�DAB ��CBA Supplements of ��s are �.

∆ABC is isosceles. Def. of isosceles ∆

�CAB ��DBA Base �s of isosceles triangles are �.

∆ABC �∆BAD ASA

DA �CB Corresponding parts of � triangles are �.

D C

B QA

E

P


UNIT 10 LESSON 2A

CUT-AND-PASTE PROOFS NAME __________________________

Directions: Cut out the steps of the proof below. Each statement and reason shouldbe a separate cutout. Challenge a partner to reconstruct the proof.

Given: MNOQ is a parallelogram.R is the midpoint of QO.

Prove: ∆MRQ � ∆PRO

Statements Reasons

MNOQ is a parallelogram. GivenR is the midpoint of QO.

QM || ON Opposite sides of a parallelogram are ||.

�Q ��POR If lines are ||, then the alternate interior �sare �.

QR � RO Definition of midpoint

�QRM � �ORP Vertical angles are �.

∆MRQ � ∆PRO ASA


UNIT 10 LESSON 2B



Q

M

R

P

O

N

Given: AB is tangent to both circles at point T.CA and BD are tangents.

Prove: CA || BD

Statements Reasons

AB is tangent to both circles at point T. GivenCA and BD are tangents.

�BTD � �ATC Vertical �s are �.

1/2 mTD = m�BTD m� formed by tangent and secant = 1

/2

1/2 mTC = m�ATC intercepted arc.

1/2 mTD = 1

/2 mTC Substitution

mTD = mTC Multiplication

m�BDT = 1/2 mTD m� formed by tangent and secant =1

/2

m�ACT = 1/2 mTC intercepted arc.

m�BDT = m�ACT Substitution

�BDT � �ACT Equal �s are congruent.

CA || BD If two lines are cut by a transversal so thatalternate interior �s are �, the lines are ||.

UNIT 10 LESSON 2C



D B

CA

T




Given: �PAD @ �QBC, AE @ BEProve: DA @ CB

Statements Reasons

1. �PAD � �QBC, AE � BE Given

2. AB � AB Reflexive Property

3. �PAD and �DAB form a straight �. Definition of straight angle�QBC and �CBA form a straight �.

4. �PAD and �DAB are supplementary. If two �s form a straight �, they are supple-�QBC and �CBA are supplementary. mentary.

5. �DAB � �CBA Supplements of � �s are �.

6. ∆ABC is isosceles Def. of isosceles ∆

7. �CAB � �DBA Base �s of isosceles triangles are �.

8. ∆ABC � ∆BAD ASA

9. DA � CB Corresponding parts of � triangles are �.

D C

B QA

E

P

LESSON 2A

Possible answer. The order of the proof has been numbered.

LESSON 1

1. Activity 1: I am an isosceles trapezoid (solvable at step 4).2. Activity 2: I am a square (solvable at step 4).3. Activity 3: I am a regular hexagon (solvable at step 5).4. Activity 4: I am an isosceles right triangle (solvable at step 3).5. Activity 5: Riddles will vary.



Given: MNOQ is a parallelogram.R is the midpoint of QO.

Prove: ∆MRQ � ∆PRO

Statements Reasons

1. MNOQ is a parallelogram. GivenR is the midpoint of QO.

2. QM || ON Opposite sides of a parallelogram are ||.

3. �Q � �POR If lines are|| , then the alternate interior �sare �.

4. QR � RO Definition of midpoint

5. �QRM � �ORP Vertical angles are �.

6. ∆MRQ � ∆PRO ASA

LESSON 2B


Q

M

R

P

O

N

Given: AB is tangent to both circles at point T.CA and BD are tangents.

Prove: CA || BD

Statements Reasons

1. AB is tangent to both circles at point T. GivenCA and BD are tangents.

2. �BTD � �ATC Vertical �s are �.

3. 1/2 mTD = m�BTD m� formed by tangent and secant = 1

/2

1/2 mTC = m�ATC intercepted arc.

4. 1/2 mTD = 1

/2 mTC Substitution

5. mTD = mTC Multiplication

6. m�BDT = 1/2 mTD m� formed by tangent and secant = 1

/2

m�ACT = 1/2 mTC intercepted arc.

7. m�BDT = m�ACT Substitution

8. �BDT � �ACT Equal �s are congruent.

9. CA || BD If two lines are cut by a transversal so that alternate interior �s are �, the lines are ||.



LESSON 2C


D B

CA

T


BRAIN TEASERS

Geoboard.....102-111

Tangrams.....112-114

Math Blaster Geometry Game Connections....115-131

1 – Building of Truth

2 – Capitol Building

3 – Cube Match

4 – Roll and Graph

5 – Transformation Game

Constructions.....132-135

World Wide Web Sites .....136-137


GEOBOARD

ACTIVITIES 1 AND 2

Tessellation is the art of tile design. Patterns occur-ring in a tessellation are of interest both artisticallyand mathematically. In mathematics, tessellation isthe repeated use of a closed figure to completelyfill in a plane with no holes or gaps. A geoboardcan be used to create many different tessellations.On the computer or using the paper geoboard,students will draw the following tessellations:

Square Tessellation – a mosaic pattern usingsquares. Hand out copies of the Square Tessellation activity sheet, p. 103.

Original Shape Tessellation – a mosaic pattern using a shape designed by the student. Hand out copies of the Original Shape Tessellation activity sheet, p. 103.

ACTIVITY 3

Hand out copies of the Dot-to-Dot activitysheets pp. 104–105. The students will draw asimple dot-to-dot drawing on the coordinateplane. They will record the ordered pairs on thesecond page and give it to a partner. Withoutlooking at the original drawing, the partner willtry to duplicate it by plotting the ordered pairs ona coordinate plane.

ACTIVITY 4

Hand out copies of the Radicals activity sheet,p. 106. Using the computer geoboard or thepaper geoboard, students will try to draw linesegments with lengths equivalent to each of theradicals. Three are impossible to draw on ageoboard.

ACTIVITY 5

Hand out copies of the Square Areas activitysheet, p. 107. Using the computer geoboard orthe paper geoboard, students will try to drawsquares having the areas listed. Three are impos-sible to draw on a geoboard.

ACTIVITIES 6 – 13

Hand out copies of the Reflections activitysheets, pp. 108–110. Students will use the papergeoboards in all activities. In Activity 6, the stu-dents will draw a polygon. Following the instruc-tions in Activities 7–13, the students will makechanges to the ordered pairs and plot the newpoints to create reflections of the original polygon.


GEOBOARD ACTIVITIES 1 AND 2

NAME __________________________

ACTIVITY 1 – SQUARE TESSELLATIONDirections: On the computer or using the paper geoboard below, tessellate the plane with squares.

ACTIVITY 2 – ORIGINAL SHAPE TESSELLATIONDirections: On the computer or using the paper geoboard below, tessellate the plane withyour own original shape.


GEOBOARD ACTIVITY 3

DOT-TO-DOT NAME __________________________

Part 1Directions: On the computer geoboard or on this worksheet, create a simple dot-to-dotdrawing. On the second sheet, make a list of the ordered pairs for each point in consecu-tive order. Give the list of ordered pairs to a partner. Ask your partner to duplicate yourdrawing without looking at the original. Be creative! Use approximately 60 points.

y

x1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1


GEOBOARD ACTIVITY 3

DOT-TO-DOT NAME __________________________Part 2

Directions: Plot the ordered pairs. See if you recreate your partner’s drawing byconnecting the points in order.

1. ( ___ , ___ ) 21. ( ___ , ___ ) 41. ( ___ , ___ )

2. ( ___ , ___ ) 22. ( ___ , ___ ) 42. ( ___ , ___ )

3. ( ___ , ___ ) 23. ( ___ , ___ ) 43. ( ___ , ___ )

4. ( ___ , ___ ) 24. ( ___ , ___ ) 44. ( ___ , ___ )

5. ( ___ , ___ ) 25. ( ___ , ___ ) 45. ( ___ , ___ )

6. ( ___ , ___ ) 26. ( ___ , ___ ) 46. ( ___ , ___ )

7. ( ___ , ___ ) 27. ( ___ , ___ ) 47. ( ___ , ___ )

8. ( ___ , ___ ) 28. ( ___ , ___ ) 48. ( ___ , ___ )

9. ( ___ , ___ ) 29. ( ___ , ___ ) 49. ( ___ , ___ )

10. ( ___ , ___ ) 30. ( ___ , ___ ) 50. ( ___ , ___ )

11. ( ___ , ___ ) 31. ( ___ , ___ ) 51. ( ___ , ___ )

12. ( ___ , ___ ) 32. ( ___ , ___ ) 52. ( ___ , ___ )

13. ( ___ , ___ ) 33. ( ___ , ___ ) 53. ( ___ , ___ )

14. ( ___ , ___ ) 34. ( ___ , ___ ) 54. ( ___ , ___ )

15. ( ___ , ___ ) 35. ( ___ , ___ ) 55. ( ___ , ___ )

16. ( ___ , ___ ) 36. ( ___ , ___ ) 56. ( ___ , ___ )

17. ( ___ , ___ ) 37. ( ___ , ___ ) 57. ( ___ , ___ )

18. ( ___ , ___ ) 38. ( ___ , ___ ) 58. ( ___ , ___ )

19. ( ___ , ___ ) 39. ( ___ , ___ ) 59. ( ___ , ___ )

20. ( ___ , ___ ) 40. ( ___ , ___ ) 60. ( ___ , ___ )


GEOBOARD ACTIVITY 4

RADICALS NAME __________________________

Directions: Use the computer geoboard or the following paper geoboards to trydrawing segments of the following lengths. Which three of these segments areimpossible to draw on a geoboard? Circle them.

�1, �2, �3, �4, �5, �6, �7, �8, �9, �10

GEOBOARD ACTIVITY 5

SQUARE AREAS NAME __________________________

Directions: Use the computer geoboard or the following paper geoboards to trydrawing squares with the following areas. Which three of these squares are impos-sible to draw on a geoboard? Circle them.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (square units)


GEOBOARD ACTIVITIES 6 AND 7


ACTIVITY 6Directions: Draw an X–Y coordinate axis dividing the paper geoboard below intofour quadrants of (roughly) equal size. Number each axis. Draw a polygon having10 or fewer sides. List the ordered pairs for each vertex of the polygon.


1. ( ___ , ___ )

2. ( ___ , ___ )

3. ( ___ , ___ )

4. ( ___ , ___ )

5. ( ___ , ___ )

6. ( ___ , ___ )

7. ( ___ , ___ )

8. ( ___ , ___ )

9. ( ___ , ___ )

10. ( ___ , ___ )

1. ( ___ , ___ )

2. ( ___ , ___ )

3. ( ___ , ___ )

4. ( ___ , ___ )

5. ( ___ , ___ )

6. ( ___ , ___ )

7. ( ___ , ___ )

8. ( ___ , ___ )

9. ( ___ , ___ )

10. ( ___ , ___ )

ACTIVITY 7Directions: Rewrite the ordered pairs using the original Y values and the oppositesof the original X values. Graph the new figure.

GEOBOARD ACTIVITIES 8 – 10


ACTIVITY 8Directions: Describe how the figure in Activity 7 compares with the original polygon.

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

ACTIVITY 9Directions: Rewrite the ordered pairs using the original X values and the oppositesof the original Y values. Graph the new figure.


1. ( ___ , ___ )

2. ( ___ , ___ )

3. ( ___ , ___ )

4. ( ___ , ___ )

5. ( ___ , ___ )

6. ( ___ , ___ )

7. ( ___ , ___ )

8. ( ___ , ___ )

9. ( ___ , ___ )

10. ( ___ , ___ )

ACTIVITY 10Directions: Describe how the figure in Activity 9 compares with the original polygonand with the polygon in Activity 7.

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________


GEOBOARD ACTIVITIES 11 – 13


ACTIVITY 11Directions: Rewrite the ordered pairs using the opposites of the original X values andthe opposites of the original Y values. Graph the new figure.

ACTIVITY 12Directions: Describe how the figure in Activity 11 compares with the original polygon.

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

ACTIVITY 13Directions: Describe what you would tell someone to do to duplicate the reflection inActivity 11. What would you call the figure?

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

1. ( ___ , ___ )

2. ( ___ , ___ )

3. ( ___ , ___ )

4. ( ___ , ___ )

5. ( ___ , ___ )

6. ( ___ , ___ )

7. ( ___ , ___ )

8. ( ___ , ___ )

9. ( ___ , ___ )

10. ( ___ , ___ )

ACTIVITIES 1 – 3 ANDACTIVITIES 6 – 13

Answers will vary.

ACTIVITY 4

Note: �3 ,�6 , and �7 are impossible todraw on a geoboard.

ACTIVITY 5

Note: 3, 6, and 7 are impossible to draw on a geoboard.


GEOBOARD – ANSWER KEY

�1 �2

8

10

9

5�4 �5

�8 �9

�10

1 2

4


TANGRAMS ACTIVITY 1

TANGRAM PIECES NAME __________________________

Directions: Tangram is an ancient Chinese game that requires these seven shapes: five triangles, one square, and one rhombus. Cut out the tangrams below. Then use them tosolve the puzzles on the following page.


TANGRAMS ACTIVITY 2

TANGRAM PICTURES NAME __________________________

Directions: Use all seven tangrams for each puzzle. Challenge yourself to constructthe magician’s hat and the rabbit. Then use your tangrams to make your own puz-zle. Can your friends solve your tangram puzzle?

TANGRAMS – ANSWER KEY


Magician’s Hat

Rabbit


MATH BLASTER GEOMETRY GAME CONNECTIONS

BUILDING OF TRUTH

OBJECTIVE: Groups of 2 to 4 players answer questions about properties of plane figures and solids.

MATERIALS:Game Board, p. 116Question Cards, pp. 117–118 (Choose one of the two sets.)Objects to use as game pieces (1 for each player)

PROCEDURE:

1. Make copies of the Game Board sheet and Question Cards for each group of players. There are two sets of question cards. Players can choose a set of cards about quadrilaterals or a set about plane figures and solids.

2. Cut out the cards, shuffle them, and place them face down in a pile.

3. Students play in a clockwise direction. The first player takes the top card and gives it to the next player to read. The first player must answer the question on the card.

4. If the answer is correct, the first player moves the number of spaces indicated on thequestion card. If the answer is incorrect, the player does not move.

5. The next player then takes a turn. Set cards aside after they are used. Players always give the card to the next player to read. Continue until a player reaches the finish line.


MATH BLASTER GEOMETRY GAME CONNECTIONSBUILDING OF TRUTH GAME BOARD

Go Back1 Space

Go

Ba

ck

1 Sp

ac

e

Go

Ba

ck

2 S

pa

ce

s

Go Back

2 Spaces

Move Ahead1 space

Mo

ve

Ah

ead

1 sp

ace

Mo

ve

Ah

ead

2 sp

ac

es

Move

Ahead2 spaces

Start

Start

Fin

ish

Fin

ish


MATH BLASTER GEOMETRY GAME CONNECTIONSBUILDING OF TRUTH QUADRILATERALS

Is a squarealways a rectangle?

(Yes - 2 spaces)

Is a

parallelogram

always a

rectangle?

(No - 4 spaces)

Can a square

be a rhombus?

(Yes - 3 spaces)

Can a

rhombus be a

parallelogram?

(Yes - 3 spaces)

Can a trape-

zoid have 4

right angles?

(No - 2 spaces)

Can a rhombus

have any sides

that are not con-

gruent?

(No - 2 spaces)

Is a square

always a

rhombus?

(Yes - 2 spaces)

Can oppositeangles of a parallelogramnot be congru-ent?

(No - 6 spaces)

Do the diago-nals of aparallelogrambisect eachother?

(Yes - 5 spaces)

Is a rectangle

always a

parallelogram?

(Yes - 2 spaces)

Can a parallelo-gram be both asquare and a rectangle at thesame time?

(Yes - 3 spaces)

Is a rhombus

always a

square?

(No - 4 spaces)

Is a rectangle

always a rhom-

bus?

(No - 2 spaces)

Is a

quadrilateral

always a

parallelogram?

(No - 4 spaces)

Are the diago-

nals of a square

perpendicular to

each other?

(Yes - 6 spaces)

Does a parallelo-gram alwayshave oppositeangles that are congruent?

(Yes - 2 spaces)

Can a trapezoid

have all sides of

different length?

(Yes - 3 spaces)

Can a trapezoid

have 2 congru-

ent sides?

(Yes - 4 spaces)

Can a

rectangle

sometimes be a

square?

(Yes - 3 spaces)

Can a square

not be a

parallelogram?

(No - 4 spaces)

Are the oppositeangles of atrapezoidalwayscongruent?

(No - 5 spaces)

Is a rhombus

always a

parallelogram?

(Yes - 3 spaces)

Can a trapezoid

have 4 congru-

ent sides?

(No - 4 spaces)

Can a trapezoid

ever be a

parallelogram?

(No - 4 spaces)

Is a parallelo-

gram always a

square?

(No - 5 spaces)

Can a trapezoid

ever have any

right angles?

(Yes - 6 spaces)

Does a trape-zoid have apair of parallelsides?

(Yes - 5 spaces)

Does a squarealways have 4congruentangles?

(Yes - 5 spaces)

Can a parallelo-

gram be a

trapezoid?

(No - 6 spaces)

Does aparallelogramalways have 4congruentangles?

(No - 2 spaces)

Must a

rhombus have

4 right angles?

(No - 3 spaces)

Can a rectangle

ever have any

obtuse angles?

(No - 4 spaces)

Must the diago-

nals of a

rhombus bisect

each other?

(Yes - 3 spaces)

Is a square

always a

parallelogram?

(Yes - 6 spaces)

Can a rhombus

be a rectangle?

(Yes - 4 spaces)

Is a parallelo-

gram always

either a square

or a rectangle?

(No - 5 spaces)


MATH BLASTER GEOMETRY GAME CONNECTIONSBUILDING OF TRUTH PLANE FIGURES AND SOLIDS

Can a scalene

triangle have

any obtuse

angles?

(Yes - 3 spaces)

If 2 angles of atriangle are con-gruent to 2angles of anothertriangle, are theysimilar?(Yes - 4 spaces)

Are all squares

similar?

(Yes - 4 spaces)

Are congruent

figures similar?

(Yes - 6 spaces)

Can an equilat-

eral triangle

have any right

angles?

(No - 5 spaces)

Can two similar

polygons have

angles with dif-

ferent measures?

(No - 4 spaces)

Can similar

polygons be

congruent?

(Yes - 2 spaces)

Does a prism

have congruent

bases?

(Yes - 5 spaces)

Is the ratio of thecircumference tothe diameter of acircle always thesame?

(Yes - 2 spaces)

Are all circles

similar figures?

(Yes - 2 spaces)

Is the sum of theangles of aquadrilateralalways the samemeasure?

(Yes - 3 spaces)

Can a triangle

have more than

one median?

(Yes - 2 spaces)

Is volume

always

measured in

cubic units?

(Yes - 4 spaces)

Are correspond-ing sides of similar polygonsalways in proportion?

(Yes - 3 spaces)

Can a pyramid

have a square

base?

(Yes - 3 spaces)

Can a parallelo-

gram and

rectangle have

the same area?

(Yes - 3 spaces)

Does a chord

always pass

through the cen-

ter of a circle?

(No - 3 spaces)

Can a right tri-

angle have 2

acute angles?

(Yes - 2 spaces)

Can a cylinderand cone withthe same baseand height havethe same volume?

(No - 2 spaces)

Is a cube always

a prism?

(Yes - 4 spaces)

Can an acute tri-

angle have a

right angle?

(No - 3 spaces)

Can a major arc

have a measure

greater than a

semicircle?

(Yes - 4 spaces)

Are area and

surface area

measured in the

same units?

(Yes - 6 spaces)

Can a minor arc

have a measure

greater than

180°?

(No - 5 spaces)

Can a scalene

triangle be

equiangular?

(No - 5 spaces)

Can a prism and

pyramid ever

have the same-

sized bases?

(Yes - 6 spaces)

Are the bases of

a prism always

parallel?

(Yes - 2 spaces)

Is area always

measured in

square units?

(Yes - 3 spaces)

Will a pyramidand prism withthe same baseand height havedifferent volumes?

(Yes - 4 spaces)

Can a triangle

have more than

one obtuse

angle?

(No - 4 spaces)

Can a pyramid

have more than

1 base?

(No - 5 spaces)

Can a pyramid

ever have all

faces the same

shape?

(Yes - 2 spaces)

Do cones and

cylinders always

have the same-

shaped bases?

(Yes - 4 spaces)

Does a pyramid

always have tri-

angular faces?

(Yes - 5 spaces)

Can an isosceles

triangle have a

right angle?

(Yes - 5 spaces)

Is the perimeter

of a rectangle

equal to 4s?

(No - 6 spaces)



CAPITOL BUILDING

OBJECTIVE: Pairs of students work together to create two different matching games. Pairs exchange games with other pairs.

MATERIALS:Covers sheet, p. 120Game Pieces sheet, p. 121

POSSIBLE TOPICS FOR GAMES:Definitions, properties, and theorems about trianglesDefinitions, properties, and theorems about quadrilateralsDefinitions, properties, and theorems about circlesDefinitions, properties, and theorems about parallel lines cut by a transversalDefinitions, properties, and theorems about angles

PROCEDURE:

1. Make copies of the Covers and Game Pieces activity sheets, pp.120 and 121.

2. Cut out the game pieces and make pairs of cards for various concepts. Forexample, one card might have the words “Pythagorean Theorem” and be matchedwith a card that says, “In a right triangle, a2 + b2 = c2.”

3. Cut out the Covers. Place a cover on top of each game card to hide it completely.

4. To play, students put the covered game cards in rows. They take turns looking for matching pairs by uncovering two cards. If a player finds a match, the player keeps both cards and takes another turn. If the two cards don’t match, they are covered upagain. Play until all cards have been matched.



CAPITOL BUILDING – COVERS



CAPITOL BUILDING – GAME PIECES



CUBE MATCH

OBJECTIVE: To match each patterned cube with the appropriate patterned net. Students can playindividually or in small groups.

MATERIALS:Cube Match sheets, pp. 123–124Cube Match Template sheets, pp. 125–126Watch or timer

PROCEDURE:

1. Make a copy of the Cube Match 1 or Cube Match 2 sheet for each student.Before students cut out the cards, have them turn the sheet over and label each matchingpair with letters or numbers so that they can check that their matches are correct when they are done.

2. Have students cut out all the cards, shuffle them, and sort them face up, with the net cards in one pile and the cube cards in another pile.

3. Players will choose one of the net cards, then find a cube card that matches the net.

4. Players should place matching cards aside, and continue until all cards have been matched.

5. Use a watch or timer to see how long it takes students to match net cards with the appropriate solids. Note: The matches in Cube Match 1 are somewhat easier than those in Cube Match 2.

6. You or your students can use copies of the Cube Match Template sheets to makeyour own matching games using patterns or colors.



CUBE MATCH 1



CUBE MATCH 2



CUBE MATCH TEMPLATE 1



CUBE MATCH TEMPLATE 2



ROLL AND GRAPH

OBJECTIVE: Students use a pair of numbers to write an equation, then graph the line for the equation. Students can work in pairs.

MATERIALS:Coordinate Plane sheet, p. 1282 number cubes (each with sides labeled 1–6 or, if available, one cube labeled 1–6 and the other labeled 4–9)

PROCEDURE:

1. Make a copy of the Coordinate Plane sheet, p.128, for each player.

2. Students take turns rolling both number cubes. One cube represents a value for x andthe other a value for y. They use the numbers rolled to write an equation. For example, after rolling a 2 and 4, a player could write the equation y = 2x or y = x + 2, where x = 2 and y = 4.

3. Now the player should find another pair of numbers that fits the equation, then plot the points and graph the line for the equation. Have students check each other’s work.

4. If a player rolls a number combination that has already been used, he or she should tryto find a different equation that will work. If no other equation is possible, the playershould roll again.

5. Continue until each player has graphed four lines. Have the class examine their graphs and discuss ideas such as slope and intersecting lines.



COORDINATE PLANE NAME __________________________

1 2 3 4 5 6 7 8 9 10 11 120-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

-11

-12

12

11

10

9

8

7

6

5

4

3

2

1



TRANSFORMATION GAME

OBJECTIVE: Students plot points to form a polygon, then construct a transformational image ofthat polygon. Students can play in pairs. Each player will create 4 figures and 4 transformation images. Students may want to use colored pencils.

MATERIALS: Coordinate Plane sheet, p. 128Transformation Game Cards sheet, p. 130Colored pencils

PROCEDURE:1. Make a copy of the Coordinate Plane sheet, p. 128, for each player. Make a

copy of the Transformation Game Cards sheet, p. 130, for each pair of players.

2. Have students cut out the cards and put them face down in separate piles, one for coordinates and one for transformations. Have pairs take turns picking up a card with coordinates and a card describing a transformation.

3. Students will plot their coordinates and join the points with line segments.

4. Next, students should draw a transformation image as described on the card.

5. Have students check each other’s work. Continue until all cards are gone.



TRANSFORMATION GAME CARDS NAME __________________________

Directions: Cut out the cards. Place the Transformation Cards in one pile and theCoordinate Pairs in another pile.

TransformationCard

HorizontalReflection

TransformationCard

Vertical Reflection

TransformationCard

Horizontal andVertical

Translation

TransformationCard

90° Rotation

TransformationCard

HorizontalTranslation

TransformationCard

VerticalTranslation

TransformationCard

180° Rotation

TransformationCard

270° Rotation

Coordinate Pairs

A(1,1), B(4,1),C(1,4)

Coordinate Pairs

L(-1,4,), M(-6,6),

N(-6,2), O(-5,1),

K(-3,1)

Coordinate Pairs

P(-2,-1), Q(-6,-1),

R(-5,-3), S(-3,-3)

Coordinate Pairs

P(1,-5), Q(1,-7),

R(6,-7), S(4,-5),

T(2,-6)

Coordinate Pairs

N(2,-1), O(4,-2),

L(5,-4), M(5,-1)

Coordinate Pairs

D(3,6), E(7,6),

G(4,9), F(8,9)

Coordinate Pairs

H(-2,6), I(-2,8),

J(-4,7)

Coordinate Pairs

T(-6,-4), U(-6,-3),

V(-10,-3), W(-10,-6),

X(-8,-6), Y(-8,-4)


GAME CONNECTIONS – ANSWER KEY

1 2 3 4 5 6 7 8 9 10 11 120-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

-11

-12

12

11

10

9

8

7

6

5

4

3

2

1

W

V

X

Y T

UR S

PQ

P

Q

TS

R

L

N MO

C

A B

D E

G F

JI

HM

NO K

L

These are the figures created from the coordinate pairs given in the Transformation Game.


CONSTRUCTIONS ACTIVITY 1

CHALLENGE! CONSTRUCT ANGLES NAME __________________________

Directions: Use what you know about perpendicular lines, angle bisectors, equiangular triangles, and duplicating an angle to construct these angles. Challenge yourself by not using a protractor to help you.

1. Construct an angle that measures 135°.

2. Construct an angle with a measurement of 22.5°.

3. Construct a 15° angle.

4. Construct an angle whose measure is 75°.



GET THE POINT(S) NAME __________________________OF CONCURRENCY? – PART 1

Directions: For each problem, construct the point of concurrency and label the point X. Then, determine any possible special characteristics of the point.

1. The incenter of a triangle is the point at which all of the angle bisectors of the triangle intersect. Construct the incenter of this triangle.

The incenter is the intersection of the three angle bisectors. What is another character-istic of the incenter? ________________________________________________________

a. Measure the distance from the point of concurrency to the three vertices.

AX = ____________ BX = ____________ CX = ____________

b. Measure the distance from the point of concurrency to the three sides.

X to BC = ____________ X to AC = ____________ X to AB = ____________

c. What is another characteristic of an incenter? __________________________________

________________________________________________________________________

REFERENCE• concurrent lines – two or more lines that intersect at the same point• point of concurrency – the point at which two or more lines intersect• four special points of concurrency in a triangle:

incenter , c i r cumcenter , or thocenter , and centro id

These points of concurrency can lie either inside or outside a triangle.

A

C

B



GET THE POINT(S) NAME __________________________OF CONCURRENCY? – PART 2

2. The circumcenter of a triangle is the point at which all of the perpendicular bisectorsof the sides of the triangle intersect. Construct the circumcenter of this triangle.

The circumcenter is the intersection of the three perpendicular bisectors of the sides ofthe triangle. What is another characteristic of the circumcenter? __________________

__________________________________________________________________________

a. Measure the distance from the point of concurrency to the three vertices.

AX = ____________ BX = ____________ CX = ____________

b. Measure the distance from the point of concurrency to the three sides.

X to BC = ____________ X to AC = ____________ X to AB = ____________

c. What is another characteristic of a circumcenter? ____________________________

__________________________________________________________________________

A

C

B



GET THE POINT(S) NAME __________________________OF CONCURRENCY? – PARTS 3–4

3. The orthocenter of a triangle is the point at which all of the altitudes of the triangle intersect. Construct the orthocenter of this triangle.

What do you notice about the orthocenter? ____________________________________

________________________________________________________________________

4. The centroid of a triangle is the point at which all of the medians of the triangle intersect. Construct the centroid of this triangle.

What are some other characteristics of the centroid? ____________________________

__________________________________________________________________________

A

C

B

A

C

B


WORLD WIDE WEB SITES TEACHER RESOURCES

These sites on the World Wide Web may be helpful when you are looking for ideasfor teaching or learning mathematics, or for professional development.

Ask Dr. Math archive of mathematical problems suitable for varying grade levelshttp://forum.swarthmore.edu/dr.math/dr-math.html

Course Materials from the Geometry Center Teacher's Guide to Building an Icosahedron as a Class Projecthttp://www.geom.umn.edu/docs/education

Davidson Softwarehttp://www.davd.com/ orhttp://www.education.com/

Departments of Ed on the Webhttp://ideanet.doe.state.in.us/htmls/states.html

Eisenhower Math Projectshttp://cesme.utm.edu/Eisenhower/Math.html

ERIC Query Formhttp://ericir.syr.edu/Eric/

Family Mathhttp://theory.lcs.mit.edu/~emjordan/FamMath.html

The Geometry Center Web Pagehttp://www.geom.umn.edu/

Geometry Forum K12 Resourceshttp://forum.swarthmore.edu/geometry/k12.geometry.html


WORLD WIDE WEB SITES TEACHER RESOURCES

CONTINUED

Internet Public Libraryhttp://www.ipl.org/ref/RR/SCI/mathematics-rr.html

Mathematical Association of America Onlinehttp://www.maa.org/

Mathematical Resources on the Webhttp://www.math.montana.edu/OtherMathSources.html

Mathematics Archives WWW Serverhttp://archives.math.utk.edu/

National Council of Teachers of Mathematics Home Pagehttp://www.nctm.org/

National Science Foundationhttp://stis.nsf.gov/

Professional Societieshttp://archives.math.utk.edu/societies.html

Unsolved Mathematics Problemshttp://www.mathsoft.com/asolve/index.html

Untangling the Mathematics of Knotshttp://www.c3.lanl.gov:80/mega-math/workbk/knot/knactiv.html

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