Unit 2

Basic Geometric Elements

Lesson 2.1

Points, Lines, and Planes

Lesson 2.1 Objectives

Define and write notation of the following: (G1.1.6) Point Line Plane Ray Line segment Collinear Coplanar End point Initial point Opposite rays Intersection

Start-Up

Give your definition of the following: Point Line

These terms are actually said to be undefined, or have no formal definition.However, it is important to have a general agreement on what each word means.

Point

A point has no dimension, it is merely a location. Meaning it takes up no space.

It is usually represented as a dot.When labeling we designate a capital letter as a name for that point. We may call it Point A.

A

Line

A line extends in one dimension. Meaning it goes straight in either a vertical,

horizontal, or slanted fashion.

It extends forever in two directions.It is represented by a line with an arrow on each end.When labeling, we use lower-case letters to name the line.

Or the line can be named using two points that are on the line.

So we say Line n, or AB

A

Bn

Plane

A plane extends in two dimensions. Meaning it stretches in a vertical direction as well as

a horizontal direction at the same time.

It also extends forever.It is usually represented by a shape like a tabletop or a wall.When labeling we use a bold face capital letter to name the plane.

Plane M Or the plane can be named by picking three points in

the plane and saying Plane ABC.

A

B

C

M

Collinear

The prefix co- means the same, or to share.Linear meansline.

A B C

We say that points A, B, and C are collinear.

So collinear means that points lie on the same line.

Coplanar

Coplanar points are points that lie on the same plane.

A

B

C

M

So points A, B, and C are said to be coplanar.

Line Segment

Consider the line AB. It can be broken into smaller pieces

by merely chopping the arrows off.

This creates a line segment or segment that consists of endpoints A and B. This is symbolized as

A

B

AB

Ray

A ray consists of an initial point where the figure begins and then continues in one direction forever. It looks like an arrow.

This is symbolized by writing its initial point first and then naming any other point on the ray, . Or we can say ray AB.

AB

A

B

Betweenness

When three points lie on a line, we can say that one of them is between the other two. This is only true if all three points are

collinear. We would say that B is between A and

C.

A B C

Opposite Rays

If C is between A and B on a line, then ray CA and ray CB are opposite rays. Opposite rays are only opposite if

they are collinear.

A BC

Intersections ofLines and Planes

Two or more geometric figures intersect if they have one or more points in common.

If there is no point or points shown, they the figures do not intersect.

The intersection of the figures is the set of points the figures have in common.

Two lines intersect at one point.Two planes intersect at one line.

A

m n

Example 2.1

Draw the following1.

2.

3.

4.

5.

6.

AB88888888888888

CD:8888888888888 8

EF

Plane DEF

intersected by at point .DE FG H

If is between and ,

draw the opposite rays and .

M N L

MN ML8888888888888888888888888888

AB

CD

FE

D EF

D

EF

G

N M L

H

Example 2.2

Answer the following1. Name 3 points that are collinear.

1. C, B, D

2. Name 3 points that are not collinear.2. ex: A, B, E or A, B, C

3. Name 3 points that are coplanar.3. ex: A, B, E or B, C, D or B, C, E

4. Name 4 points that are not coplanar.4. ex: A, B, E, C

5. What are two ways to name the plane?5. Plane ABE or Plane F

6. What are two names for the line that passes through points C and B.

6. line g orBC:8888888888888 8

Homework 2.1

Lesson 2.1 – Point, Line, Plane p1-2

Due Tomorrow

Lesson 2.2

Distance,Midpoint, andSegment Addition

Lesson 2.2 Objectives

Utilize the distance formula. (G1.1.3)

Apply the midpoint formula. (G1.1.5)

Justify the construction of a midpoint. (G1.1.5)

Utilize the segment addition postulate. (G1.1.3)

Identify the symbol and definition of congruent. (G1.1.3)

Define segment bisector. (G1.1.3)

Postulate 1: Ruler Postulate

The points on a line can be matched to real numbers called coordinates.The distance between the points, say A and B, is the absolute value of the difference of the coordinates. Distance is always positive.

A BCE D

Length

Finding the distance between points A and B is written as AB

Writing AB is also called the length of line segment AB.

Postulate 2: Segment Addition Postulate

If B is between A and C, then AB + BC = AC.

Also, the opposite is true. If AB + BC = AC, then B is between A

and C.

A B C

AB BC

AC

Example 2.3

1. Sketch and write the segment addition postulate if point E is between points D and F. D FE

DE + EF = DF

2. Sketch and write the segment addition postulate if point M is between points N and P.

N PM

NM+ MP= NP

Example 2.4Find1. GJ

1. GJ = 16

2. KM2. KM = 36

3. XY3. 71-29

XY = 424. LM

4. x + 2x = 183x = 18x = 6LM = 6

Distance Formula

To find the distance on a graph between two points

A(1,2) B(7,10) We use the Distance Formula

(x2 – x1)2 + (y2 – y1)2AB =

Distance can also be found using the Segment Addition Postulate, which simply adds up each segment of a line to find the total length of the line.

Congruent Segments

Segments that have the same length are called congruent segments. This is symbolized by .

If you want to state two segments are congruent, then you write

If you want to state two lengths are equal, then you write

Hint: If the symbols are there, the congruent sign should be there.

NTLE NTLE

Example 2.5

Find the distance of each segment and identify if any of the segments are congruent.

1. J(1,1)K(0,5)

2 2(0 1) (5 1)

2. L(2,1)M(-2,0)

3. A(4,3)B(-1,6)

4. D(2,-3)E(-2,0)

2 2( 1) (4)

1 1617 4.12

2 2( 2 2) (0 1) 2 2( 4) ( 1)

16 117 4.12

2 2( 2 2) (0 3) 2 2( 4) (3)

16 925 5

2 2( 1 4) (6 3) 2 2( 5) (3)

25 934 5.83JK LM

(x2 – x1)2 + (y2 – y1)2

Midpoint

The midpoint of a segment is the point that divides the segment into two congruent segments. The midpoint bisects the segment, because

bisect means to divide into two equal parts.

J YOWe say that O is the midpointof line segment JY.

Midpoint Formula

A(1,2) B(7,10)

We can also find the midpoint of segment AB byusing its endpoints in…

The Midpoint Formula

Midpoint of AB = (x1 + x2)

(y1 + y2)( ),2 2

This gives the coordinates of the midpoint, or point that is halfway between A and B.

Example 2.6

Find the midpoint1. R(3,1)

S(3,7)2. T(2,4)

S(6,6)

( ),(x1 + x2) 2

(y1 + y2) 2

3 3 1 7,

2 2

6 8,

2 2

3,4

2 6 4 6,

2 2

8 10,

2 2

4,5

Finding the Other End

Many may say finding the midpoint is easy!

It is simply the average of the two endpoints.

Now imagine knowing the midpoint, one endpoint, and trying to find the coordinates of the other endpoint.

Try to remember what the midpoint formula does and work it backwards.

So here is what we are going to do:

1. Double the coordinates of the midpoint.

2. Subtract the coordinates of the known endpoint.

(3,1)E

E

M

?

(5,7)M 2

(10,14)

(3,1)(7,13)

(7,13)F

Example 2.7

Find the other endpoint given one endpoint, E, and the midpoint, M.

1. E(0,5)M(3,3)

2. E(-1,-3)M(5,9)

(3,3) 2(6,6)(0,5)(6,1)

(5,9) 2(10,18)( 1, 3) (11,21)

Segment Bisector

A segment bisector is a segment, ray, line, or plane that intersects the original segment at its midpoint.

J YO

H

T

segment bSo is a of isec .torHT JY:8888888888888 8

Example 2.8

Use the diagram to find the given measure if line l is a segment bisector.

109 in

ST = ½(109 in)

ST = 54.5 in

Homework 2.2

Lesson 2.2 – Line Segments p3-4

Due Tomorrow

Lesson 2.3

Angles and Their Measures

Lesson 2.3 Objectives

Identify more than one name for an angle. (G1.1.6)

Identify angle measures. (G1.1.6)

Classify angles as right, obtuse, acute, or straight. (G1.1.6)

Apply the angle addition postulate. (G1.1.3)

Utilize angle vocabulary to solve problems. (G1.1.6)

Define angle bisector and its uses. (G1.1.3)

What is an Angle?

An angle consists of two different rays that have the same initial point.The rays form the sides of the angle.The initial point is called the vertex of the angle. Vertex can often be thought of as a

corner.

Naming an Angle

All angles are named by using three points First, name a point that lies on one side of the angle. Second, name the vertex next.

The vertex is always named in the middle. Finally, name a point that lies on the opposite side

of the angle.

W

O N

So we can call It WONOr NOW

Using a ProtractorTo measure an angle with a protractor, do the following:

1. Place the cross-hairs of the protractor on the vertex of the angle.2. Line up one side of the angle with the 0o line near the bottom of the

protractor.3. Read the protractor for the where the other side of the angle points.

54o

Example 2.9

Protractor Stations

Congruent Angles

Congruent angles are angles that have the same measure. To show that we are finding the measure of

an angle… Place a “m” before the name of the angle.

Equal Measures

Congruent Angles

NOWmWONm NOWWON

Types of Angles

Acute Right Obtuse Straight

Looks like

MeasureLess than

90(<90)

Equal to 90(=90)

Greater than 90

(>90)

Equal to 180(=180)

Example 2.10Give another name for the angle in the diagram above. Then, tell

whether the angle appears to be acute, obtuse, right, or straight.

1. JKN1. NKJ, K

1. right

2. KMN2. NMK

2. straight

3. PQM3. MQP

3. acute

4. JML4. LMJ

4. acute

5. PLK5. KLP

5. obtuse

Other Parts of an Angle

The interior of an angle is defined as the set of points that lie between the sides of the angle.The exterior of an angle is the set of points that lie outside of the sides of the angle.

Interior

Exterior

Postulate 4: Angle Addition Postulate

The Angle Addition Postulate allows us to add each smaller angle together to find the measure of a larger angle.

32o

17o

What is the total?

49o

Example 2.11

Use the given information to find the indicated measure.

1.

2.

3x + 15 + x + 7 = 94

4x + 22 = 94

4x = 72

x = 1825o

3x + 1 + 2x – 6 = 135

5x – 5 = 135

5x = 140

x = 28

85o

3(28) + 1

84 + 1

85

Adjacent Angles

Two angles are adjacent angles if they share a common vertex and side, but have no common interior points. Basically they should be touching, but

not overlapping. C

A

T

R

CAT and TAR are adjacent.

CAR and TARare not adjacent.

Angle Bisector

An angle bisector is a ray that divides an angle into two adjacent angles that are congruent. To show that angles are congruent,

we use congruence arcs.

Example 2.12In the diagram, bisects . Find .BD ABC m ABC

2. 3.

4x = 3x + 6

x = 6

1.

4(6) + 3(6) + 6

24 + 18 + 6

mABC = 48o

5x – 11 = 4x + 1

x – 11 = 1

5(12) – 11 + 4(12) + 1

60 – 11 + 48 + 1

mABC = 98o

8x – 16 = 4x + 20

4x – 16 = 20

2(4(9) + 20)

mABC = 112o

x = 124x = 36

x = 9

2(56)

Homework 2.3

Lesson 2.3 - Angles and Their Measures p5-7

Due Tomorrow

Lesson 2.4

Angle Pair Relationships

Lesson 2.4 Objectives

Identify vertical angle pairs. (G1.1.1)

Identify linear pairs. (G1.1.1)

Differentiate between complementary and supplementary angles. (G1.1.1)

Vertical Angles

Two angles are vertical angles if their sides form two pairs of opposite rays.

Basically the two lines that form the angles are straight.

To identify the vertical angles, simply look straight across the intersection to find the angle pair.

Hint: The angle pairs do not have to be vertical in position.

Vertical Angle pairs are always congruent!1

324

1

324

Linear Pair

Two adjacent angles form a linear pair if their non-common sides are opposite rays.

Simply put, these are two angles that share a straight line.

Just like neighbors share a fence line, but they must live on the same side of the road.

Since they share a straight line, their sum is… 180o

1 2

Example 2.13

Find the measure of all unknown angles, when m1 = 57o.

1. m21. 123o

2. m32. 57o

3. m43. 123o

Example 2.14

Solve for x and y.1.

3x + 48 = 180

3x = 132

x = 44

2.

9x + 7 = 5x + 67

4x + 7 = 67

4x = 60

x = 15

3.

7x = 5x + 18

2x = 18

x = 9

16y - 27 = 13y

-27 = -3y

y = 9

Complementary v Supplementary

Complementary angles are two angles whose sum is 90o. Complementary

angles can be adjacent or non-adjacent.

Supplementary angles are two angles whose sum is 180o. Supplementary

angles can be adjacent or non-adjacent.

Example 2.15Find the measure of all unknown angles, given that m and n form a

right angle and the m1 = 22o and 1 4.1. m2

1. 68o

2. m52. 22o

3. m63. 68o

4. m44. 22o

5. m35. 68o

6. m76. 68o

7. m87. 22o

m

n

Example 2.16

A and B are complementary. Find mA and mB.

1. mA = 2x + 12 mB = 9x – 10

1. 2x + 12 + 9x – 10 = 9011x + 2 = 9011x = 88x = 8

mA = 2(8) + 12

mA = 16 + 12

mA = 28o

mB = 9(8) - 10

mB = 72 - 10

mB = 62o

A and B are supplementary. Find mA and mB.

2. mA = 12x + 32 mB = 4x – 12

2. 12x + 32 + 4x – 12 = 18016x + 20 = 18016x = 160x = 10mA = 12(10) + 32

mA = 120 + 32

mA = 152o

mB = 4(10) - 12

mB = 40 - 12

mB = 28o

Perpendicular Lines

When two lines intersect to form a right angle, they are said to be perpendicular lines.

a

bSo we can say that a .b

Homework 2.4

Lesson 2.4 - Angle Pair Relationships p7-8

Due Tomorrow

Lesson 2.5

Introduction to Parallel LinesandTransversals

Lesson 2.5 Objectives

Identify angle pairs formed by a transversal. (G1.1.2)

Compare parallel and skew lines. (G1.1.2)

Lines and Angle Pairs

1

5

2

8

4

6

3

7Corresponding Angles – because they lie in corresponding positions of each intersection.

TransversalAlternate Exterior Angles – because they lie outside the two lines and on opposite sides of the transversal.

Alternate Interior Angles – because they lie inside the two lines and on opposite sides of the transversal.

Consecutive Interior Angles – because they lie inside the two lines and on the same side of the transversal.

Example 2.17

Determine the relationship between the given angles1 3 and 9

1) Alternate Interior Angles

2 13 and 52) Corresponding Angles

3 4 and 103) Alternate Interior Angles

4 5 and 154) Alternate Exterior Angles

5 7 and 145) Consecutive Interior Angles

Postulate 15:Corresponding Angles Postulate

If two parallel lines are cut by a transversal, then corresponding angles are congruent.

You must know the lines are parallel in order to assume the angles are congruent.

1

5

2

8

4

6

3

7

Theorem 3.4:Alternate Interior Angles

If two parallel lines are cut by a transversal, then alternate interior angles are congruent.

Again, you must know that the lines are parallel. If you know the two lines are parallel, then identify where the

alternate interior angles are. Once you identify them, they should look congruent and they

are.

1

5

2

8

4

6

3

7

Theorem 3.5:Consecutive Interior Angles

If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary.

Again be sure that the lines are parallel. They don’t look to be congruent, so they MUST be

supplementary.

5

4

6

31 2

8 7

+=180o + = 180o

Theorem 3.6:Alternate Exterior Angles

If two parallel lines are cut by a transversal, then alternate exterior angles are congruent.

Again be sure that the lines are parallel.

1 2

8

5

4

6

3

7

Example 2.18Find the missing angles for the following:

120o

120o

105o

105o

110o

110o 70o

120o

60o

140o

140o

Example 2.19

Solve for x1.

2.

3.

4.

3x + 15 = 60

3x = 45

x = 15

x – 10 = 100

x = 110

2x – 4 = 92

2x = 96

x = 48

5x – 10 + 75 = 180

5x + 65 = 180

5x = 115

x = 23

Parallel versus Skew

Two lines are parallel if they are coplanar and do not intersect.

The short-hand symbol for being parallel is //.

Lines that are not coplanar and do not intersect are called skew lines. These are lines that look like they intersect

but do not lie on the same piece of paper.

Skew lines go in different directions while parallel lines go in the same direction.

Example 2.20

Complete the following statements using the words parallel, skew, perpendicular.

1) Line WZ and line XY are _________.1) parallel

2) Line WZ and line QW are ________.2) perpendicular

3) Line SY and line WX are _________.3) skew

4) Plane WQR and plane SYT are _________.4) parallel

5) Plane RQT and plane WQR are _________.5) perpendicular

6) Line TS and line ZY are __________.6) skew

7) Line WX and plane SYZ are __________.7) parallel.

Homework 2.5

Lesson 2.5 p11-12

Due TomorrowUnit 2 Test Monday, October 11th