40 Block 2 ~ Lines And Angles ~ Triangles And Quadrilaterals w ord w All BLOCK 2 ~ LINES AND ANGLES TRIANGLES AND QUADRILATERALS LESSON 7 CLASSIFYING TRIANGLES ----------------------------------------------- 42 Explore! Naming By Sides LESSON 8 ANGLE SUM OF A TRIANGLE -------------------------------------------- 47 Explore! Add Them Up LESSON 9 SPECIAL TRIANGLES ---------------------------------------------------- 52 Explore! What Makes Me Special? LESSON 10 CONGRUENT AND SIMILAR TRIANGLES ----------------------------------- 56 LESSON 11 PARALLEL LINES AND SIMILAR TRIANGLES -------------------------------- 61 Explore! Parallel Similarity LESSON 12 ANGLE SUM OF A QUADRILATERAL -------------------------------------- 66 Explore! Four Corners LESSON 13 SPECIAL QUADRILATERALS ---------------------------------------------- 71 REVIEW BLOCK 2 ~ TRIANGLES AND QUADRILATERALS ---------------------------- 76
Transcript
40 Block 2 ~ Lines And Angles ~ Triangles And Quadrilaterals
isosceles triAngle
word wAll
PArAllelogrAm
QuAdrilAterAl
scAlene triAngle
corresPonding PArts
trAPeZoid
congruent Figures
isosceles trAPeZoid
eQuiAngulAr
eQuilAterAl triAnglesimilAr Figures
BLoCK 2 ~ LInes And AngLestriangLes and QuadriLateraLs
Explore! Naming By SidesLesson 8 angLe sUM oF a TriangLe -------------------------------------------- 47
Explore! Add Them UpLesson 9 sPeciaL TriangLes ---------------------------------------------------- 52
Explore! What Makes Me Special?Lesson 10 congrUenT and siMiLar TriangLes ----------------------------------- 56
Lesson 11 ParaLLeL Lines and siMiLar TriangLes -------------------------------- 61
Explore! Parallel SimilarityLesson 12 angLe sUM oF a QUadriLaTeraL -------------------------------------- 66
Explore! Four CornersLesson 13 sPeciaL QUadriLaTeraLs ---------------------------------------------- 71
review BLock 2 ~ TriangLes and QUadriLaTeraLs ---------------------------- 76
Block 2 ~ Triangles And Quadrilaterals ~ Tic - Tac - Toe 41
BLoCK 2 ~ trIAngLes And QuAdrILAterALs
tic - tac - tOehigh school choices
Investigate math courses off ered at the high school
you will attend.
See page for details.
exterior Angles
Discover and apply a unique property
in all triangles.
See page for details.
triAngle moBile
Create an original piece of art showing the
properties of triangles.
See page for details.
lAnd oF sPeciAl Figures
Write a fi ction story about the Land of Special Figures.
Include triangles and quadrilaterals in the story.
See page for details.
sum oF interior Angles
Discover the rule for the sum of the interior angles
in any polygon.
See page for details.
constructing segments
Use a compass and straightedge to duplicate and combine segments.
See page for details.
triAngle ineQuAlity theoreM
Determine the possible measurements for the third
side of a given triangle.
See page for details.
clAssiFicAtion gAme
Make a memory card game where players classify triangles by
sides and angles.
See page for details.
too tAll to Know
Find the height of objects that are too tall to measure
with a measuring tape.
See page for details.
42 Lesson 7 ~ Classifying Triangles
A triangle is a polygon with three sides. A triangle can be classifi ed by its angles and by the lengths of its sides. In Block 1, you learned about four types of angles: acute, obtuse, right and straight. When a triangle is classifi ed by its angles, these same words are used. Since a triangle cannot have a straight angle, there are three classifi cations of triangles by angles.
Acute triangles have three acute angles
right triangles have one right angle
obtuse triangles have one obtuse angle
In the last block, angles were marked as congruent using tick marks on the arcs. In the same way, tick marks can show that the lengths of two line segments are equal.
|| ||
| |||||||
cLassiFying triangLes
Lesson 7
Lesson 7 ~ Classifying Triangles 43
step 1: Examine each group of triangles in the table above. List any similarities you see in each category.
step 2: Identify how each group is different from the other groups. Specifically, identify why triangles from one group do not belong in the other group.
step 3: Angles can be classified by the lengths of their sides. Triangles can be equilateral, scalene or isosceles. Based on your observations of the triangles in the table above, write a definition for each type of triangle.
step 4: Sketch your own triangle for each group.
All sides of an equilateral triangle are the same length. An isosceles triangle has at least two sides with the same measure. A scalene triangle has no sides with the same measure. If a triangle is a scalene triangle, each side of the triangle will be a different length.
Classify each triangle by its sides and angle measures.a. b. c.
a. Two sides are the same length so it is an isosceles triangle. One angle in the triangle is more than 90°, so it is an isosceles, obtuse triangle.b. No sides are the same length. There is a right angle. The triangle is a scalene, right triangle.c. All angles in the triangle are less than 90° and all sides are the same length. It is an acute, equilateral triangle.
sketch a diagram to represent an acute, isosceles triangle named ∆ABC.
In order to be an acute triangle all angles must be less than 90°. Since the triangle is isosceles, at least two sides must be the same length. Tick marks can be used to show which sides are equal lengths in the sketch.
exercises
1. What are the three triangle classifications based on side lengths?
2. What are the three triangle classifications based on angle measures?
Classify each triangle by its angle measures.
3. 4. 5.
6. 7. 8.
Classify each triangle by its side lengths.
9. 10. 11.
exampLe 2
solution
| |
A
B
C
40°
50° 60°
60° 60°
30°
|
60°
|
|
|18°
18°
115°42°
23°
7
7
76.4 m
2 m
7.1 m ||||
75°|
Lesson 7 ~ Classifying Triangles 45
12. 13. 14.
Classify each triangle by its sides and angle measures.
15. 16.
17. 18.
19. A sail on a sailboat has one 90° angle and the sides are three different lengths. Classify this triangle by its sides and angle measures.
20. Another sail on the sailboat is an acute triangle with side lengths of 10 feet, 6 feet and 10 feet. Classify this sail by its sides and angle measures.
draw and label each triangle to match each description.
The side lengths of a triangle are given. Classify the triangle by its sides.
27. 1, 1, 1 28. 8, 8, 10 29. 6, 8, 10
30. h, h, h 31. 5p, 3p, 5p 32. 3j, 4j, 5j
33. Can a right triangle have more than one right angle? Support your answer with a diagram.
34. In an obtuse triangle one angle is obtuse. What type of angle are the other two angles?
60°
60° 60°
||
|
6 ft
4 ft 4 ft
|||
|||
|
|
|
||||
38°
71° 71°
2 cm
3 cm4 cm
|||
|||117°
46 Lesson 7 ~ Classifying Triangles
review
solve for x. Check your solution. 35. 36.
37. 38.
(20x + 11)°
(15x + 36)°
>>
>>
(9x + 1)°125°
>>>>
(6x − 3)°111°
5x°
(3x + 60)°
tic-tAc-toe ~ cl As siF icAtion gA me
Create a memory card game where players must match cards with triangles with given angle measures, side lengths and/or diagrams to other cards that classify the triangle. You can include some cards that classify only by sides or angles and others that classify by both sides and angles. Make sure each information card has a classifi cation card to match it. Make a set of at least twelve pairs of cards. Try playing the game with a classmate. If it does not work, make the needed adjustments before turning in the game.
Example of a matching set:
Create a memory card game where players must match cards with triangles
ScaleneRight
Triangle8
6
10
Lesson 8 ~ Angle Sum Of A Triangle 47
The sum of the measures of the angles of every triangle is the same. In this lesson you will determine the angle sum of a triangle. Th is property of triangles will be very useful as you apply it to triangles in many situations.
step 1: Draw the three triangles listed below on a blank sheet of paper. Make the triangles large enough to measure their angles with a protractor.
right triangleAcute triangle
obtuse triangle
step 2: Use a protractor to measure each angle in all three triangles. Write the measure of each angle inside the triangle.
step 3: Find the sum of the angles in each triangle.
step 4: Do you notice any similarities in the sums of the angles in each triangle? If possible, write a rule for the sum of the measures of the angles of any triangle.
step 5: Compare your triangle sums and rule with a classmate. Did he/she get the same or similar results?
Th e sum of the angles of a triangle can also be shown using the method below. A triangle is drawn on a piece of paper and cut out. Th e angles are torn apart and lined up. Th e three angles form a straight angle. Th e measure of a straight angle is 180°.
expLOre! add them up
angLe sum OF a triangLe
Lesson 8
1
3
2 23121 3
180°
48 Lesson 8 ~ Angle Sum Of A Triangle
set up an equation and solve for x.
The sum of the angles in a triangle is 180°. 80 + 65 + x = 180 Combine like terms. 145 + x = 180 Subtract 145 from each side of the equation. −145 −145 x = 35 The measure of the missing angle is 35°.
∆You has the angle measures listed below. m∠Y = 70° m∠o = (3x – 10)° m∠u = 7x°
a. set up an equation. solve for x. b. Find the degree measure of each angle.
a. The angles of a triangle sum to 180°. 70 + (3x – 10) + 7x = 180 Combine like terms. 10x + 60 = 180 Subtract 60 from each side of the equation. −60 −60 Divide both sides of the equation by 10. 10x ___ 10 = 120 ___ 10
x = 12
b. Write the given expression for each angle. m∠O = (3x − 10)° m∠U = 7x Substitute 12 for x. = 3(12) − 10 = 7(12) Multiply. = 36 − 10 = 84 Subtract. = 26° m∠O = 26° m∠U = 84° ☑ m∠Y + m∠O + m∠U = 180° 70° + 26° + 84° ?= 180° 180° = 180°
exampLe 2
solutions
exampLe 1
solution
65°
80°
x°
Lesson 8 ~ Angle Sum Of A Triangle 49
exercises
Find the degree of each missing angle.
1. 2. 3.
4. 5. 6.
set up an equation and solve for x.
7. 8. 9.
10. 11. 12.
13. Use ∆AMT at the right. a. Set up an equation to find the value of x. b. Solve for x. c. Find the measure of each angle.
14. The m∠C = 60°, m∠U = (7 + 5x)° and m∠P = (1 + 3x)° in ∆CUP. a. Set up an equation and solve for x. b. Find the m∠C, m∠U and m∠P.
120°
x°
25°
39°
x° 51°
x°
76°
|| ||
x°45°x°
12°
147° x°
99°45°
5x°
4x°
(x + 5)°
72°
29°x°
x° x°
10x°
x°
x°2x°(3x +5)°
6x°
(2x + 1)°
(3x + 3)°
2x°
(x − 6)°T
A
M
50 Lesson 8 ~ Angle Sum Of A Triangle
15. ∆PRT is an isosceles triangle. The measure of ∠P is (2x + 1)°. The other two angles each measure 42°. a. Set up an equation and solve for x. b. Find m∠P.
16. Jeff determined that the value of x in the triangle at the right is 11. a. Find the value of each angle by substituting 11 for x. b. Was Jeff ’s solution of x = 11 correct? How do you know?
17. Siena determined that x = −8 in the triangle at the left. a. Find the value of each angle by substituting −8 for x. b. Was Siena’s solution of x = −8 correct? How do you know? c. Find the correct value of x.
review
Classify each triangle by its sides and angles.
18. 19. 20.
Fill in each blank with the appropriate word or number.
21. Same-side interior angles add up to ______ degrees when between parallel lines.
22. Alternate interior angles are _________ to each other when between parallel lines.
23. A ___________ is the line that cuts through a set of parallel lines.
24. Complementary angles add up to ______ degrees.
25. An angle that equals 180° is called a ______________ angle.
7x°
(4x + 2)°
(5x + 2)°
B
A
C
(5 − 4x)°(17 − 3x)°
(50 − 5x)°
S
L
K
|| 107°
|| |||
49°
72°59°|
|
|
Lesson 8 ~ Angle Sum Of A Triangle 51
tic-tAc-toe ~ exte r ior Angle s
Th e sum of the remote interior angles in any triangle is congruent to the measure of the corresponding exterior angle. Below is a diagram showing the remote interior angles and the corresponding exterior angle.
An algebraic proof of the exterior angle and remote interior angles relationship shows that the sum of the remote interior angles equals the measure of the corresponding exterior angle.
statement reason a + b + c = 180° Th e sum of the angles of a triangle is 180°. c + d = 180° Angles c and d are supplementary. a + b + c = c + d Substitute c + d for 180°. −c −c Subtract c from both sides. a + b = d
solve for x.
1. 2. 3.
4. One of the remote interior angles is 66°. Th e exterior angle is a right angle. What is the degree measure of the other remote interior angle?
5. Th e exterior angle measures 78°. Give a possible pair of degree measures that the remote interior angles could be.
solve for x. Find the measure of each angle inside the triangle.
6. 7. 8.
1
2
3
b
c da
61°
42°
x°
x°
133° 111°
2x°
50°138°
3x°
91°(5x − 5)°
B
CA
(x − 62)°
1 _ 2 x°
x°
S
R T(12 − 4x)°
(1 − 2x)°
67°
N
M P
52 Lesson 9 ~ Special Triangles
You have classifi ed triangles as equilateral, isosceles or scalene depending on the lengths of their sides. When a triangle has two or more sides that are the same length, the angles in that triangle have unique properties. Complete the Explore! below to discover these properties.
step 1: Two equilateral triangles are drawn below. Measure the angles inside each triangle and list them on your own paper.
step 2: Based on Lesson 8, what should the sum of the angle measures of each triangle equal? step 3: Is the sum of ∠A, ∠B and ∠C equal to 180°? If not, check your measurements. Is the sum of ∠D, ∠E and ∠F equal to 180? If not, check your measurements.
step 4: Do you notice anything about the measure of each angle in an equilateral triangle? If so, what is your discovery?
step 5: Use division to show how you could calculate the degree measure of an angle in an equilateral triangle.
step 6: Use a ruler to draw two isosceles triangles. Remember that two sides must be the same length in an isosceles triangle.
step 7: Measure the angles in your triangles. Th ere should be two angles in each triangle that are equal to each other. Where are those angles in comparison to the two sides that are equal?
expLOre! what makes me speciaL?
speciaL triangLes
Lesson 9
A
B
C
D
E
F
Lesson 9 ~ Special Triangles 53
Equilateral triangles are also equiangular. Equiangular means that all angles have the same measure.
∆MnP is an equilateral triangle. Th e measure of ∠M is (2x + 6)°. Find the value of x. Each angle in an equilateral triangle is equal to 60°. Set the angle equal to 60°. 2x + 6 = 60Subtract 6 from each side of the equation. −6 −6Divide both sides of the equation by 2. 2x __ 2 = 54 __ 2 Th e value of x is 27. x = 27
Find the value of x in the diagram below.
Th e triangle above is isosceles. Th e angles that are across from the congruence marks must also be equal so:
Th e sum of the angles in a triangle is 180°. x + x + 112 = 180Combine like terms. 2x + 112 = 180 Subtract 112 from both sides of the equation. −112 −112Divide both sides of the equation by 2. 2x __ 2 = 68 __ 2 Th e missing angles in the diagram are 34°. x = 34
exampLe 1
solution
60°
60°60°|||
||| |||
||| |||
||
exampLe 2
solution
x°
112°||
x°
112°||
x°
54 Lesson 9 ~ Special Triangles
exercises
1. What is the measure of each angle in an equilateral triangle?
2. Which two angles in an isosceles triangle are equal? Draw a diagram to illustrate your answer.
Find the value of x in each diagram.
3. 4. 5.
6. 7. 8.
9. ∆YUM has two angles that measure 78°. a. Sketch a diagram of ∆YUM. b. Find the measure of the other angle. c. Classify ∆YUM by its sides and angles.
10. All three sides in ∆BET are 2 inches in length. a. Sketch a diagram of ∆BET. b. What is the degree measure of the angles? c. Classify ∆BET by its sides and angles.
11. ∆JAM is isosceles and has an angle measuring 35°. Sketch two possible diagrams of ∆JAM with the angle measures labeled. 12. One angle in an equilateral triangle is (8x − 24)°. Solve for x.
13. Hayley’s house makes an isosceles triangle with her school and the mall as shown in the diagram below.
a. Find the value of x. b. How far is it from Hayley’s house to the mall?
71° x°
||
|
|
(x + 24)°
2x°
||
|(x +3)°
60°
60°
60°
3x − 5
10
x°
34°55
5x°
|
|
Hayley’s House
MallSchool
10 − 4x2 miles
5 miles40° 40°
Lesson 9 ~ Special Triangles 55
14. Hans designed a photo frame shaped like an isosceles triangle. He wanted two of the sides of the frame to each be three times the length of the shortest side. a. Hans made the shortest side of the triangle 4 inches long. Sketch a diagram of his photo frame.
b. Th e angle across from the shortest side of the frame is 20°. Determine the measures of the other two angles. Explain your reasoning.
review
solve for x in each triangle. 15. 16. 17.
18. Th e complement of ∠B is 37°. Find m∠B.
19. Th e supplement of ∠L is 146°. Find m∠L.
20. Two angles in a triangle are 18° and 102°. Find the measure of the third angle in the triangle.
21. One angle in a linear pair is 85°. Find the measure of the other angle in the linear pair.
22. Use the diagram at the right. a. Find m∠1. b. Find m∠2. c. Find m∠3. d. Find m∠4.
x°
49°
23°
(10x − 1)°
23°
(x + 10)°(2x − 20)°
40°
tic-tAc-toe ~ tr i A ngle moBile
A mobile is a type of art, oft en referred to as kinetic art. It hangs in space and uses balance and motion. A mobile may be more commonly recognized hanging above a baby lying in their crib. Create a mobile that shows all of the diff erent things you learned about triangles in this block. Use defi nitions, rules and properties about triangles as information on the pieces of your mobile.
98°
123
4<
<
56 Lesson 10 ~ Congruent And Similar Triangles
Two triangles that are the exact same shape and the exact same size are called congruent fi gures. Two triangles that have the exact same shape, but not necessarily the exact same size are called similar fi gures. Th e parts of the fi gures that correspond are called corresponding parts. Look at the two similar triangles below.
Th e corresponding angles in similar triangles are congruent. Th e corresponding sides are proportional. You can show two triangles are similar using the similar symbol: ∆CAT ~ ∆DOG.
∆MnP is similar to ∆JKL. Find the value of x, y and z.
Since the triangles are similar, the corresponding m∠L = m∠P angles are congruent. m∠L = x = 50
Th e sides of congruent triangles are proportional. NP ___ KL = MP ___ JL
Substitute the values of each known side into the 12 __ 8 = y __ 10 proportion.
Set the cross products equal to each other. 8y = 120Divide both sides of the equation by 8. y = 15
To fi nd the value of z, set the sum of the three angles z + 50 + 90 = 180in the triangle equal to 180°. z + 140 = 180Subtract 140 from both sides of the equation. −140 −140 z = 40
Corresponding Angles Corresponding sides∠C and ∠D
___ CT and
___ DG
∠A and ∠O ___
TA and ___
GO ∠T and ∠G
___AC and
___ OD
exampLe 1
solution
M P
NK
J L
812
yz° x°
1050°
cOngruent and simiLar triangLes
Lesson 10
D
O
35°
120° 25° G8
612
T
C
A
35°
120° 25°
6
3
4
Lesson 10 ~ Congruent And Similar Triangles 57
If two angles in one triangle are congruent to two angles in another triangle, then the two triangles are similar. Th is rule is referred to as the Angle-Angle Similarity Rule.
Th e triangles shown above are similar because ∠A ≅ ∠P and ∠N ≅ ∠I. Th e Angle-Angle Similarity Rule can be proven by showing that if two angles are congruent in the triangles, the third angle is also congruent.
Since all three angles in the triangles are congruent, the triangles are similar.
In previous books you learned about slope triangles. You were able to choose any two points on a line and form a slope triangle.
On the graph at the left , a right angle is formed in each slope triangle.
Th e corresponding sides of the triangles are parallel because they are either both vertical or both horizontal.
Since the lines are parallel, the blue line is the transversal. As you learned in Block 1, this makes the top angles in the triangles corresponding angles. Th erefore, those angles are congruent. Th is is also true of the bottom angles.
Since two angles in the triangles are congruent, the slope triangles are similar.
N
131°
27°A T
131°27°P
I
G
58 Lesson 10 ~ Congruent And Similar Triangles
exercises
Find the corresponding sides and corresponding angles to the ones given for each pair of fi gures. 1. 2.
a. ____
MN corresponds to ____ ∠M ≅ ∠___ a. ___
TA corresponds to ____ ∠M ≅ ∠___ b.
___ AN corresponds to ____ ∠N ≅ ∠___ b.
___ AR corresponds to ____ ∠O ≅ ∠___
c. ____
MA corresponds to ____ ∠A ≅ ∠___ c. ___
TR corresponds to ____ ∠P ≅ ∠___
3. Sketch a similar triangle to the right triangle below. Include angle and side lengths in your drawing.
4. Explain the diff erence between congruent triangles and similar triangles.
Find the missing side length(s) for each set of similar fi gures.
5. 6.
7. 8.
9. 10.
M 2.
NA
12 14
1860°
80°
40°
T
O P
6 7
960°
80°40°
M
O P
159
1240°
50°
5
4
3
36°
54°
10
20 30
x x
14
35
24.5
x 20
25
9 12
1520
22
8
4xy
10
10
xy12 15.4
50°
50°
T
A R
3 5
440°
50°
30
18 20
y x
60
Lesson 10 ~ Congruent And Similar Triangles 59
11. Which set of triangles in exercises 5 through 10 are congruent figures? Explain your reasoning.
12. Are the triangles below similar? Explain your answer.
13. Are the triangles below similar? Explain your answer.
14. Which triangles below are similar to ∆RAP? Explain your reasoning for each.
15. Explain how two slope triangles formed on the same line are similar. Draw a diagram to support your answer.
16. Sketch a pair of triangles that are congruent. Label all the angles and sides.
17. Draw two triangles of different sizes that are similar to ∆MQT. Label all angle measures and side lengths.
85°
30°
85°
30°
6
10 8
8
15 12
60°
80° 40°
6 10
8R P
A
60°
80°
E
T
P64
5
R
LG
20 16
12
N
MA
50°
70°
A
C
T
98°
52°8
10
5
M
Q
T
60 Lesson 10 ~ Congruent And Similar Triangles
review
name the special angle relationship between the two angles. Find x. 18. 19. 20.
21. Two angles are supplementary. Th ey measure (5x – 4)° and (9x + 16)°. a. Write an equation and solve for x. b. Find the measure of each angle.
98°
(4x − 2)°
(x + 9)°
75°(x + 10)°
(2x − 20)°
tic-tAc-toe ~ too tA l l to Know
Very tall objects are diffi cult to measure. Th is activity requires that you go outside to fi nd the height of objects that cannot be measured using traditional means. It is important that all measurements are made within about one hour total time. It must be sunny to complete this activity.
step 1: Begin by measuring your height in feet. Write the inches part of the measurement as a fraction over 12.
step 2: Go outside and measure the length of your shadow in feet.
step 3: At the same time as you measure your own shadow, measure the shadow of at least 5 objects such as a fl agpole, the goal posts on a football fi eld, a tree or a building. Measure in feet with remaining inches written as a fraction of a foot.
step 4: Draw yourself as a stick-fi gure with your shadow on the ground. In the drawing, connect the end of the shadow to the top of the stick fi gure’s head. Th is should make a right triangle. Label all known lengths.
step 5: Draw triangles for each object you measured just as you did in step 4. Th ese triangles are similar to your stick fi gure triangle. Why is this true?
step 6: Find the height of each object using the similar triangles.
step 7: Explain why it was important to measure your shadow and the other object’s shadows at the same time of day.
Very tall objects are diffi cult to measure. Th is activity requires that you
Lesson 11 ~ Parallel Lines And Similar Triangles 61
In the fi rst block of this book you learned about parallel lines, transversals and special angle pairs. In this lesson, you will use some of these special angle pairs. Similar triangles can be formed by sets of parallel lines and two transversals that intersect one another.
step 1: Copy or trace the diagram at the right. Label the points and angles as shown.
step 2: Angles 1 and 3 make what special angle pair?
step 3: What type of special angle pair are ∠4 and ∠2?
step 4: What is true about the pairs of angles in steps 2 and 3?
step 5: What can you conclude about ∆ABC and ∆DEC? Why?
step 6: Copy or trace the diagram to the right. Explain how it is the same type of problem as the diagram shown above.
step 7: What angle is shared by ∆STU and ∆RTV?
step 8: Find the measure of ∠STU.
step 9: What is the measure of ∠TVR?
step 10: Because the angles in ∆STU and ∆RTV are congruent, the triangles are similar. Find the value of x using a proportion.
As you have seen in the Explore!, special angle pairs can be useful when trying to determine if two triangles are similar. Once you have shown that two angles in a triangle are equal to two angles in another triangle, then you know the triangles are similar based on the Angle-Angle Similarity Rule.
expLOre! paraLLeL simiLarity
A
3
B1 2
D E
C
4>>
>>
>>
>>
65
4
50°
x
55°S
T
U
R V
paraLLeL Lines and simiLar triangLes
Lesson 11
62 Lesson 11 ~ Parallel Lines And Similar Triangles
show that ∆JKL ~ ∆MnL.
∠LMN ≅ ∠LJK because they are alternate interior angles.
∠LNM ≅ ∠LKJ because they are alternate interior angles.
∆JKL ~ ∆MNL because two angles in each triangle are congruent to one another. Th is is based on the Angle-Angle Similarity Rule.
Find the missing measures in the two similar triangles.a. m∠hAeb. m∠Yc. x
a. Corresponding angles are congruent. m∠T = m∠HAE Substitute 58° for ∠T. 58° = m∠HAE
b. Th e sum of three angles in a triangle is 180°. m∠T + m∠H + m∠Y = 180° Substitute the angle values for ∠T and ∠H. 58° + 77º + m∠Y = 180° Combine like terms. 135º + m∠Y = 180° Subtract 135 from both sides of the equation. m∠Y = 45°
c. Write a proportion with corresponding sides. AE ___ TY = HE ___ HY Fill in the known lengths. x __ 21 = 6 __ 14
Set the cross products equal to each other. 126 = 14x Divide by 14 on both sides of the equation. 126 ___ 14 = 14x ___ 14
9 = x
exampLe 1
solution
J
N
L
M
K
>>
>>
exampLe 2
solutions
>>
>>
H
77°
E
6
8
Y21
Ax
T 58°
J
N
L
M
K
<<
<< |
|||
||
Lesson 11 ~ Parallel Lines And Similar Triangles 63
exercises
1. Use the diagram at the right. a. What special angle pair do ∠JLK and ∠LMN represent? Are they congruent angles? b. Choose the correct word to complete the statement: ∠JKL and ∠KNM are congruent or supplementary. c. Complete the statement: ∆LJK ~ ∆_____
2. Use the diagram at the left. a. Name the two pairs of angles that are congruent inside the triangles based on alternate interior angles. b. Complete the statement: ∆POW ~ ∆_____
3. Use the diagram at the right. a. What angle in ∆AND measures 56°? b. What is the m∠AND? c. What is the m∠AYS? d. How do you know ∆AND ~ ∆AYS?
4. Use the diagram at the left. a. What angle besides ∠CBD measures 37°? b. What is the measure of ∠CDB? c. What is the measure of ∠C? d. Find the value of x.
5. Sketch two similar triangles using parallel lines and transversals. Use congruence marks to show the angles that are congruent.
Find the values of x and y in each pair of similar triangles. 6. 7.
8. 9.
N >>
>>K
J
L
M
R>
>2OP
W
1
E4 3
>D>
N
A
S
Y
56°
48°
>>
>>
D
C
E61°
37°8
x
6B
9A
27
9
40°
21
x°
y} >
>
>
>x° y°
52°
37°
> x°y°
39° 71°>
70°12 8
4
x°>>
>>
40°
75°
y°
64 Lesson 11 ~ Parallel Lines And Similar Triangles
Find the measure of a, b, c, d and e.
10. 11.
12. Sketch similar triangles formed by parallel lines, transversals and alternate interior angles.a. Label the vertex, or corner, of each triangle with a letter.b. Identify the corresponding sides. c. Identify the corresponding angles.
review
13. Use the triangle at the right.a. Write an equation and solve for x.b. Find the measure of ∠H.c. Find the measure of ∠G.
14. Th e beams forming the roof of a house form an isosceles triangle. Copy the diagram at right and fi ll in the missing angles.
15. An angle in an equilateral triangle measures (3x − 60)°. Find the value of x.
>
>>>
>>
>>39°
2124
a° 76°14
c°
e
d
42
b°
2x°
(x + 12)°
4
3
5
6.25
a
b
tic-tAc-toe ~ high school choice s
Investigate the course off erings in mathematics at the high school you will be (or are) attending by looking at their course catalog or on their website. What courses are off ered? How many of the courses are you required to take to graduate? Is there a sequence of courses that students must follow? What level of math must you reach to be eligible for entry into a four-year university?
Create a brochure illustrating your fi ndings. Survey your friends to see if they have any questions about high school mathematics. If the information is not already covered in your brochure, try to locate the answer. Include the answers to their questions in a “Frequently Asked Questions” section on the back of your brochure.
c°
e°
d°
37°
F H
G
|
35°
|
Lesson 11 ~ Parallel Lines And Similar Triangles 65
tic-tAc-toe ~ constructi ng segm e nts
Use a compass and straightedge to duplicate segments and make combinations of segments. Use the segments below.
1. Duplicate ___
AB . a. Use a straightedge to draw a segment longer than
___ AB .
b. Measure ___
AB by placing the stylus on A and opening the compass so the pencil is on B. c. Using this setting, place the stylus on one endpoint of your segment then make a mark that crosses your segment. d. Label your congruent segment
___ XY .
2. Duplicate ___
GH .
3. Construct ___
EF + ___
GH . a. Use a straightedge to draw a segment longer than the combined length of
___ EF and
___ GH .
b. Measure ___
EF by placing the stylus on E and opening the compass so the pencil is on F. c. Using this setting, place the stylus on one endpoint of your segment then make a mark that crosses your segment. d. Measure
___ GH by placing the stylus on G and opening the compass so the pencil is on H.
e. Using this setting, place the stylus on your intersection from part c, then make a mark that crosses your segment. Th is mark will be further down the segment making it longer. f. Label the segment
___ EH that is equal to the combined length of
___ EF and
___ GH .
4. Construct ___
AB + ___
EF .
5. Construct 2 ∙ ___
GH .
6. Construct ___
JK – ___
CD . a. Use a straightedge to draw a segment longer than
___ JK .
b. Measure ___
JK with a compass. c. Using this setting, mark off the length of
___ JK on your segment.
d. Measure ___
CD with a compass. e. Using this setting, place the stylus on the intersection from part c and turn the compass backwards to mark off the length of
___ CD . Th e length of
___ JK should be
reduced or taken away from. f. Label the segment
____ VW that represents the length of
___ JK with
___ CD removed from it.
7. Construct ___
JK – ___
GH .
8. Construct 2 ∙ ___
EF – ___
JK .
G H
A B
C D
J
E
K
F
66 Lesson 12 ~ Angle Sum Of A Quadrilateral
A quadrilateral is a polygon with four sides. In previous math classes, you have probably learned about some common quadrilaterals such as squares, rectangles, parallelograms and trapezoids. Try the Explore! to discover the sum of the angles in a quadrilateral.
step 1: Trace the parallelogram at the right.
step 2: Connect two opposite corners in the quadrilateral with a line segment. How many triangles are formed?
step 3: Th e sum of the angles in a triangle is 180°. Multiply the number of triangles formed in the parallelogram in step 2 by 180°. What is the sum of the angles in the parallelogram? step 4: Draw a square. a. What is the degree measure of one angle in a square? b. How many angles are in a square? c. Find the sum of the angles in your square. Does it match your answer in step 3?
step 5: What is the degree measure of one angle in a rectangle? What is the sum of the four angles in a rectangle?
step 6: Draw a large quadrilateral that is not a square, rectangle or parallelogram. Some examples are shown below.
step 7: Use a protractor to measure the angles in your quadrilateral. Add the four angles together.
step 8: Write a rule about the angle sum of the four angles in any quadrilateral.
expLOre! FOur cOrners
angLe sum OF a QuadriLateraL
Lesson 12
Lesson 12 ~ Angle Sum Of A Quadrilateral 67
set up an equation and solve for x.
Th e sum of the angles of a quadrilateral is 360°. 110 + 51 + 33 + x = 360 Combine like terms. 194 + x = 360 Subtract 194 from each side of the equation. −194 −194 x = 166 ☑ 110 + 51 + 33 + 166 ?= 360 360 = 360 In quadrilateral QuAd, m∠u = 2x°, m∠A = (x + 30)° , ∠u ≅ ∠d and ∠Q is a right angle.a. draw a diagram and label it.b. set up an equation and solve for x.c. Find the degree measure for each angle.
a.
b. Th e sum of the angles of a quadrilateral is 360°. 90 + 2x + (x + 30) + 2x = 360 Combine like terms. 120 + 5x = 360 Subtract 120 from each side of the equation. −120 −120 Divide both sides of the equation by 5. 5x __ 5 = 240 ___ 5 x = 48 c. To fi nd degree measures, substitute 48 for x. m∠U = m∠D = 2(48) = 96° m∠A = (48 + 30) = 78° ∠Q is a right angle. m∠Q = 90°
exampLe 1
solution
110°
51°
33°
x°
exampLe 2
solutions 2x°
2x°(x + 30)°
Q U
DA
68 Lesson 12 ~ Angle Sum Of A Quadrilateral
exercises
set up an equation and solve for x.
1. 2. 3.
4. 5. 6.
set up an equation and solve for x. Find the degree measures of each unknown angle.
7. 8. 9.
10. 11. 12.
13. A quadrilateral has angles that measure 146°, 75° and 84°. What is the measure of the missing angle?
109°x°
68°83°
148° 100°
37°
x°
85°
149°
x°
x°142°
57°
116°
x°
x°
89°
75°
x°98°
>
>3x°
2x°| |
||||
U
R S
Y
84°
40°
(4x + 9)°
(9x + 6)°
P
N
M
Q
(2x + 10)°
(3x − 30)°
C
D
B
A
(2x − 8)°
104°
165° x°
F
GH
E 5x°
4x°
54°
I
L
K
J
(3x + 10)°
2x°
5x°
4x°
V
W
Z
Y
Lesson 12 ~ Angle Sum Of A Quadrilateral 69
14. Patty owns a piece of farmland that is an irregular quadrilateral. She wants to put a fence around her farm. She needs to know the angles at each corner of her land. She measured three of the angles and found they were 85°, 120° and 50°. a. What is the angle measure of the fourth corner? b. Sketch a diagram of Patty’s land.
15. In quadrilateral JUMP, ∠J and ∠U are both 65°. The measure of ∠M is (3x + 1)° and the measure of ∠P is (2x – 6)°. a. Draw a diagram and label it. b. Set up an equation and solve for x. c. Find the degree measure of ∠M and ∠P.
16. Each angle in quadrilateral WXYZ is (4x – 11)°, (7x + 2)°, (2x + 33)° and x°, respectively. a. Draw a diagram and label it. b. Set up an equation and solve for x. c. Find the degree measure of each angle.
17. Quadrilateral GOAT has the following angle measures: m∠G = (x + 5)°, m∠O = (2x – 15)°, m∠A = 4x° and ∠T is a right angle. a. Draw a diagram and label it. b. Set up an equation and solve for x. c. Find the degree measure of each angle.
review
Find the value of x in each figure. 18. ∆RST ~ ∆MNP 19. ∆ABC ~ ∆UYZ
20. 21.
N
x
PM
6
S
5
R
4
TC
14
A B24
U 8
Z
x
Y
x°
73° 69° >>
>>
>
>
10
78
x
70 Lesson 12 ~ Angle Sum Of A Quadrilateral
tic-tAc-toe ~ sum oF inte r ior Angle s
Th e sum of the three angles in a triangle is 180°. Discover how to fi nd the sum of the angles of any polygon based on its number of sides.
1. Copy and complete the chart.
Polygon name and number of sides diagram number of triangles
sum of degree measures in the
triangles
Conclusion: Angle sum
Quadrilateral4 sides 2
180° + 180°or
2(180)°360°
Pentagon5 sides 3
hexagon6 sides
heptagon7 sides
octagon8 sides
nonagon9 sides
decagon10 sides
2. Does a pattern form in the angle sums? How does the pattern relate to the number of sides of the polygon?
3. Write a formula to calculate the degree measure of any polygon based on its number of sides, n.
4. Find the degree measure of these polygons:
a. Dodecagon (12 sides) b. 15-gon c. 24-gon d. 41-gon
Lesson 13 ~ Special Quadrilaterals 71
A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Rectangles and squares are types of parallelograms.
In all parallelograms, the opposite sides are congruent and the opposite angles are congruent.
Two consecutive angles in a parallelogram are supplementary. Each pair of consecutive angles will always add up to 180°. In the diagram above, two consecutive angles, 70° and 110°, sum to 180°.
Find the values of a, b and c.
Th e variable a is opposite the angle which measures 54°, so a = 54.
Th e variable b is a consecutive angle with 54º. Th is makes the angles supplementary. b + 54 = 180° Subtract 54 from both sides of the equation. b = 126
Th e variable c is opposite the side which measures 13 units, so c = 13.
exampLe 1
solution
>>>>
>
>
54°
a°
b°
c
13
7
> >
>>
>>70°
70°
110°
110°
speciaL QuadriLateraLs
Lesson 13
> >
>>
>>>>
>>>
>
>>>> > >
>>
>>> >
72 Lesson 13 ~ Special Quadrilaterals
A trapezoid is a quadrilateral with exactly one pair of parallel sides. A trapezoid has two bases and two legs.If the legs are congruent, it is an isosceles trapezoid.
If a trapezoid is isosceles, then each pair of base angles is congruent.
A top base angle and a bottom base angle form a pair of same-side interior angles. Th is means they are supplementary. In the example at the right: 114° + 66° = 180°
Find the values of x, y and z.
Th e variable x is congruent to the opposite leg, so x = 12.
Th e variable y forms a pair of base angles with 99°, so y = 99.
Th e angle represented by z is the supplement of 99° because the angles form a pair of one top and one bottom base angle. z + 99 = 180Subtract 99 from both sides of the equation. −99 −99 z = 81
>
>
trapezoid
Base
Base
Leg Leg Isoscelestrapezoid
>
>
| |
114° 114°
66° 66°>
>
|
exampLe 2
solution
>
>99°
18
12x
z°
y°
| |
|
Lesson 13 ~ Special Quadrilaterals 73
exercises
1. Draw a parallelogram. Write angle measures in the parallelogram so that opposite angles are equal and consecutive angles are supplementary. The sum of the angles must be 360°.
2. Draw an isosceles trapezoid. a. Place congruence marks on the appropriate sides. b. Write angle measures in the trapezoid so that each pair of base angles is congruent and pairs of top and bottom base angles are supplementary. The sum of the angles must be 360°.
Find the values of x and y in each figure. 3. 4.
5. 6.
7. 8.
9. 10.
82°>
|
>
|
y°
17
2y 6
3x + 5>>
>>
> >
>>>>
>
>y° 115°
(x + 8)°
19
>
|
>
|
2x°
x°
3 − 4y
>>>>
>
>
96°
y°
x°
(5y − 4)°
(2x − 1)°
61°
| |
>
>
2y°
15 72°
|
|
> >
10x
6
y + 4
11
(5x − 25)°
>
>
>>
(4x + 1)°
>>
x°
74 Lesson 13 ~ Special Quadrilaterals
11. Explain what special angle pair is used to determine that a top base angle and a bottom base angle in an isosceles trapezoid are supplementary.
12. Lucy and Eddie each drew a parallelogram that measured 1 inch on every side. Lucy argues that Eddie’s figure is a square, not a parallelogram. Do you agree or disagree with Lucy? Why?
13. A teacher asked her math class to draw an isosceles trapezoid with one base that is 3 centimeters long. The other base is 7 centimeters long. Their trapezoids should have a pair of 60° angles and a pair of 120° angles. Will everyone’s trapezoid look the same? Why or why not? Support your answer with sketches.
14. Find the values of a, b, c, d and e in the figure below.
15. Find the values of a, b, c and d in the figure below.
||
>
>
67°
c°
a°
6d
(10b − 3)°
18
a°
9
(2c + 6)°
>>
>>
> >
b°
4e + 3
5d − 1
21
52°
Lucy’s Drawing
Eddie’s Drawing
Lesson 13 ~ Special Quadrilaterals 75
review
Find the values of x and y in each fi gure. 16. 17.
18. 19. ∆HJK ~ ∆PGR
3x°(−5 + 5y)° 120°
113° x°
(2y − 7)°
>
>
>
>
3
2
34
y
x
H
12
J
13
K5
P
y4
G x R
tic-tAc-toe ~ lA n d oF sPeci A l Figu r e s
Write a fi ction story about the Land of Special Figures. Th e story should include the Special Triangles and Quadrilaterals from Lessons 9 and 13. Also research and include at least one additional special fi gure and its properties (i.e. rhombus, kite, pentagon, etc). Th roughout the story, the fi gures should reveal properties about their sides and angles. Title your
story and include illustrations.
76 Block 2 ~ Review
Lesson 7 ~ Classifying Triangles
Classify each triangle by its sides and angle measures. 1. 2. 3.
4. 5. 6.
sketch and label a triangle to match each description.
congruent fi gures isosceles trapezoid quadrilateral corresponding parts isosceles triangle scalene triangle equiangular parallelogram similar fi gures equilateral triangle trapezoid
vocabulary
BLoCK 2
Block 2 ~ Review 77
Lesson 8 ~ Angle Sum of a Triangle
solve for x in each triangle.
10. 11. 12.
13. Use ∆APE at the right. a. Set up an equation to find the value of x. b. Solve for x. c. Find m∠A and m∠P.
14. In ∆CAR, m∠A = (15 – 3x)° and m∠R = (5 – 2x)°. The measure of ∠C is 80°. a. Sketch and label a diagram of ∆CAR. b. Write an equation and solve for x. c. Find m∠A and m∠R.
15. Nancy determined that the value of x in the triangle at the right is 20. a. Find the measure of each angle by substituting 20 for x. b. Was Nancy’s solution of x = 20 correct? How do you know?
Lesson 9 ~ Special Triangles
Find the value of x in each diagram. 16. 17. 18.
19. ∆TWL has two angles that measure 63°. a. Sketch a diagram of ∆TWL. b. Find the measure of the third angle. c. What type of triangle is ∆TWL based on its side lengths and angle measures?
20. One angle in an equilateral triangle is (6x + 12)°. Solve for x.
(x + 2)°
3x°
A
PE
R
T S
2x °
(4x − 8)° (3x + 6)°
x°
25°
x°
46°
||
||
||
|(2x + 4)°
30°
30°x°
2x° 2x°
122°
(2x − 3)°
(4x − 3)°
||
78 Block 2 ~ Review
21. Francisco has a triangular window in his bedroom. It is the shape of an equilateral triangle. a. What are the measures of each angle in the window? b. He has equal-sized pieces of framing he wants to put around the window. It takes 6 pieces of framing to cover one side of the window. How much will it take to go around the entire window?
22. Shannon’s home makes an isosceles triangle with her school and the park as shown in the diagram below.
a. Find the value of x. b. How far is it from Shannon’s home to the park?
Lesson 10 ~ Congruent and Similar Triangles
23. ∆PLA ~ ∆NET. List the corresponding angles and sides.
solve for x in each set of similar triangles.
24. 25.
26. 27.
28. Are the triangles below similar? Explain your answer.
35°
(3.5 − 2x) km1.5 km
35°
Home
ParkSchool
58° 58°
2.5 km
T E
N
A L
P
x
20
3
5
x
4
8
6
x
4
5
2
x
1.5
12
6
Block 2 ~ Review 79
Lesson 11 ~ Parallel Lines and Similar Triangles
29. Use the diagram at the right. a. What angle in ∆WSB measures 49°? b. What is the m∠WAM? c. What is the m∠BWS? d. What property can you use to say ∆WAM ~ ∆WBS?
30. Use the diagram at the left. a. What angle besides ∠YFG measures 42°? b. What is the measure of ∠FYE? c. What is the measure of ∠M? d. Fill in the blank: ∆MYF ~ ∆____.
Find the values of x and y in each pair of similar triangles. 31. 32.
33. Find the measure of a, b, c, d and e.
34. Sketch similar triangles formed by parallel lines, transversals and same-side interior angles. Use congruence marks to show the congruent angles.
12.5
5
452°
78°b°
d
c°
a°
e°
>
>
>
>
>
10
6
6
62°
xy°
50°
>
>
8y°
56
71°
442°
x
>
>
42°F
M
Y
58° GE
>
>
M49°
W
A
B S51°
80 Block 2 ~ Review
Lesson 12 ~ Angle Sum of a Quadrilateral
35. What is the sum of the angles in a quadrilateral?
set up an equation and solve for x.
36. 37. 38.
39. Find the degree measure of the two unknown angles in ABCD.
40. A quadrilateral has one right angle, a 143° angle and a 56° angle. What is the measure of the fourth angle?
41. In quadrilateral TEND, ∠T and ∠E are both 60°. The measure of ∠N is (6x + 1)° and the measure of ∠D is (4x + 9)°. a. Draw a diagram and label it. b. Set up an equation and solve for x. c. Find the degree measure of ∠N and ∠D.
Lesson 13 ~ Special Quadrilaterals
42. Draw an isosceles trapezoid. a. Place congruence marks on the appropriate sides. b. Write angle measures in the trapezoid so that each pair of base angles is congruent and pairs of top and bottom base angles are supplementary. The sum of the angles must be 360°.
Find the value of x in each figure. 43. 44.
45. 46.
64°
102° x°
97°
10x° 110°
3x°
55°
80° (1 − 11x)°
(1 − 6x)°57°
98°
x°| |
>
>
110° 70°
10x°
>
>>
>
>>
(3x + 14)° (5x − 2)°
| |
>
>(x + 20)°
4x°
> >
>>
>>
(2x + 1)°
(5x + 7)°
118°38°A
B
C
D
Block 2 ~ Review 81
47. A skate jump is the shape of an isosceles trapezoid. Th e bottom angles of the trapezoidal jump are 40° each. What is the measure of one of the top angles of the jump?
48. Explain what special angle pair is used to determine that two consecutive angles in a parallelogram are supplementary.
tic-tAc-toe ~ tr i A ngle ineQuA l it y theor e m
Th e Triangle Inequality Th eorem states that for any triangle, the length of a given side must be less than the sum of the other two sides but greater than the diff erence between the two sides.
For example, two sides of a triangle are 3 and 8. Th e length of the third side, x, must be:
◆ Less than the sum of the given two sides (3 and 8). x < 11 ◆ Greater than the diff erence of the given two sides (3 and 8). x > 5 Th ese fi ndings can be written as a compound inequality. 5 < x < 11
Write a compound inequality showing the possible lengths of the third side of a triangle given the other two lengths.
1. 5 and 9 2. 10 and 12 3. 6 and 7
Write the possible integer side lengths for the third side of a triangle given the other two lengths.
4. 4 and 7 5. 2 and 5 6. 1 and 9
determine if each triangle has measurements that can form a triangle. If not, re-draw the fi gure with one side length changed. show all work.
7. 8. 9.
Th e Triangle Inequality Th eorem states that for any triangle, the length
9
54
12
10
3
5
3
1
82 Block 2 ~ Review
Alden Architect
PortlAnd, oregon
I am an architect. Architects are designers. Designs can be as small as a piece of furniture or as large as a city. The primary focus for most architects is buildings. Architects use a process to complete projects which includes determining what a client wants, designing it and drawing up plans. Architects also help oversee the bidding process and actual construction of a project.
Math is used everyday in architecture. Proportion, scale and dimension are important components architects use to create a plan. Architects also have to estimate how much a project will cost. If a client cannot afford to build what the architect has designed, then the design is of no use. Architects prepare budgets and cost estimates throughout the entire architectural process to avoid this problem.
Architects have to do math calculations to account for every square inch of space. Calculations have to be very precise to make sure the buildings designed will not fall over.
People need a Bachelor’s degree in Architecture from a college or university to become an architect. This usually takes about five years. After getting a degree, people do on-the-job training. They must pass tests to become a licensed architect. Architects often acquire a Master’s degree in Architecture.
The average starting salary for a college graduate architect is $35,000 - $45,000 per year depending on where you live in the country. A mid-level architect makes between $55,000 - $75,000 per year. A senior architect or a firm principal can make $80,000 - $100,000 or more.
Architecture offers something new every day. Whether it is putting together a design proposal, preparing a set of construction drawings, overseeing construction in the field or picking materials for a new project, architects are constantly being challenged with new tasks. Also, as a building goes up, an architect gets to see their hard work paying off. It is truly a rewarding profession.
