Points, Lines, Planes, and Angles

5-1

Warm Up

Problem of the Day

Lesson Presentation

Pre-Algebra

Warm Up Solve. 1. x 30 90 2. 103 x

180 3. 32 x 180 4. 90 61 x 5. x 20

90

x 60

x 77

x 148

x 29

x 70

Problem of the Day Mrs. Meyers class is having

a pizza party. Half the class wants pepperoni on

the pizza, of the class wants sausage on the

pizza, and the rest want only cheese on the

pizza. What fraction of Mrs. Meyers class wants

just cheese on the pizza?

Learn to classify and name figures.

Vocabulary

point line plane segment ray angle right

angle acute angle obtuse angle complementary

angles supplementary angles vertical

angles congruent

Points, lines, and planes are the building blocks

of geometry. Segments, rays, and angles are

defined in terms of these basic figures.

A point names a location.

A

Point A

A line is perfectly straight and extends forever

in both directions.

A plane is a perfectly flat surface that extends

forever in all directions.

E

P

plane P, or plane DEF

D

F

A segment, or line segment, is the part of a line

between two points.

H

G

A ray is a part of a line that starts at one

point and extends forever in one direction.

J

KJ

K

Additional Example 1A 1B Naming Points, Lines,

Planes, Segments, and Rays

A. Name 4 points in the figure.

Point J, point K, point L, and point M

B. Name a line in the figure.

Any 2 points on a line can be used.

Additional Example 1C Naming Points, Lines,

Planes, Segments, and Rays

C. Name a plane in the figure.

Any 3 points in the plane that form a triangle

can be used.

Additional Example 1D 1E Naming Points, Lines,

Planes, Segments, and Rays

D. Name four segments in the figure.

E. Name four rays in the figure.

Try This Example 1A 1B

A. Name 4 points in the figure.

Point A, point B, point C, and point D

B. Name a line in the figure.

Any 2 points on a line can be used.

A

B

C

D

Try This Example 1C

C. Name a plane in the figure.

Any 3 points in the plane that form a triangle

can be used.

A

B

C

D

Try This Example 1D 1E

D. Name four segments in the figure

E. Name four rays in the figure

A

B

C

D

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The measures of angles that fit together to form

a straight line, such as ?FKG, ?GKH, and ?HKJ,

add to 180.

The measures of angles that fit together to form

a complete circle, such as ?MRN, ?NRP, ?PRQ, and

?QRM, add to 360.

A right angle measures 90. An acute angle

measures less than 90. An obtuse angle measures

greater than 90 and less than 180. Complementary

angles have measures that add to 90.

Supplementary angles have measures that add to

180.

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Additional Example 2A 2B Classifying Angles

A. Name a right angle in the figure.

?TQS

B. Name two acute angles in the figure.

?TQP, ?RQS

Additional Example 2C Classifying Angles

C. Name two obtuse angles in the figure.

?SQP, ?RQT

Additional Example 2D Classifying Angles

D. Name a pair of complementary angles.

m?TQP m? RQS 47 43 90

?TQP, ?RQS

Additional Example 2E Classifying Angles

E. Name two pairs of supplementary angles.

?TQP, ?RQT

m?TQP m?RQT 47 133 180

m?SQP m?RQS 137 43 180

?SQP, ?RQS

Try This Example 2A

A. Name a right angle in the figure.

?BEC

Try This Example 2B 2C

B. Name two acute angles in the figure.

?AEB, ?CED

C. Name two obtuse angles in the figure.

?BED, ?AEC

Try This Example 2D

D. Name a pair of complementary angles.

?AEB, ?CED

m?AEB m?CED 15 75 90

Try This Example 2D 2E

E. Name two pairs of supplementary angles.

m?AEB m?BED 15 165 180

?AEB, ?BED

m?CED m?AEC 75 105 180

?CED, ?AEC

- Congruent figures have the same size and shape.
- Segments that have the same length are

congruent. - Angles that have the same measure are congruent.
- The symbol for congruence is ?, which is read

is congruent to. - Intersecting lines form two pairs of vertical

angles. Vertical angles are always congruent, as

shown in the next example.

Additional Example 3A Finding the Measure of

Vertical Angles

In the figure, ?1 and ?3 are vertical angles, and

?2 and ?4 are vertical angles.

A. If m?1 37, find m? 3.

The measures of ?1 and ?2 add to 180 because

they are supplementary, so m?2 180 37

143.

The measures of ?2 and ?3 add to 180 because

they are supplementary, so m?3 180 143

37.

Additional Example 3B Finding the Measure of

Vertical Angles

In the figure, ?1 and ?3 are vertical angles, and

?2 and ?4 are vertical angles.

B. If mÐ4 y, find mÐ2.

m?3 180 y

m?2 180 (180 y)

180 180 y

Distributive Property m?2 m?4

y

Try This Example 3A

In the figure, ?1 and ?3 are vertical angles, and

?2 and ?4 are vertical angles.

2

3

A. If m?1 42, find m?3.

1

4

The measures of ?1 and ?2 add to 180 because

they are supplementary, so m?2 180 42

138.

The measures of ?2 and ?3 add to 180 because

they are supplementary, so m?3 180 138

42.

Try This Example 3B

In the figure, ?1 and ?3 are vertical angles, and

?2 and ?4 are vertical angles.

2

3

B. If m?4 x, find m?2.

1

4

m?3 180 x

m?2 180 (180 x)

180 180 x

Distributive Property m?2 m?4

x

Lesson Quiz

In the figure, ?1 and ?3 are vertical angles, and

?2 and ?4 are vertical angles.

1. Name three points in the figure.

Possible answer A, B, and C

2. Name two lines in the figure.

3. Name a right angle in the figure.

Possible answer ?AGF

4. Name a pair of complementary angles.

Possible answer ?1 and ?2

5. If m?1 47, then find m? 3.

47