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Points, Lines, Planes, and Angles

Course 3

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

Points, Lines, Planes, and Angles

1

3

1

6

Course 3

Problem of the Day

Mrs. Meyer’s 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. Meyer’s class wants just cheese on the pizza?

Points, Lines, Planes, and Angles

Course 3

Insert Lesson Title Here

Vocabulary

point line plane

segment ray angle

right angle acute angle

obtuse angle complementary angles

supplementary angles

vertical angles

congruent

Points, Lines, Planes, and Angles

Course 3

Points, lines, and planes are the building blocks of geometry. Segments, rays, and angles are defined in terms of these basic figures.

Points, Lines, Planes, and Angles

C

l

B

line l, or BC

Course 3

A line is perfectly straight and extends forever in both directions.

Points, Lines, Planes, and Angles

Course 3

A plane is a perfectly flat surface that extends forever in all directions.

P

E

plane P, or plane DEF

D

F

Points, Lines, Planes, and Angles

GH

Course 3

A segment, or line segment, is the part of a line between two points.

H

G

Points, Lines, Planes, and Angles

Course 3

A ray is a part of a line that starts at one point and extends forever in one direction.

J

KJ

K

Points, Lines, Planes, and Angles

KL or JK

Course 3

Additional Example 1: 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.

Points, Lines, Planes, and Angles

Plane , plane JKL

Course 3

Additional Example 1: 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.

Points, Lines, Planes, and Angles

JK, KL, LM, JM

KJ, KL, JK, LK

Course 3

Additional Example 1: Naming Points, Lines, Planes, Segments, and Rays

D. Name four segments in the figure.

E. Name four rays in the figure.

Points, Lines, Planes, and Angles

BC

DA or

Course 3

Check It Out: Example 1

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.

B

A

C

D

Points, Lines, Planes, and Angles

Plane , plane ABC, plane BCD, plane CDA, or plane DAB

Course 3

Check It Out: Example 1

C. Name a plane in the figure.

Any 3 points in the plane that form a triangle can be used.

B

A

C

D

Points, Lines, Planes, and Angles

AB, BC, CD, DA

DA, AD, BC, CB

Course 3

Check It Out: Example 1

D. Name four segments in the figure

E. Name four rays in the figure

B

A

C

D

Points, Lines, Planes, and Angles

An angle () is formed by two rays with a common endpoint called the vertex (plural, vertices). Angles can be measured in degrees.

One degree, or 1°, is of a circle. m1

means the measure of 1. The angle can be named XYZ, ZYX, 1, or Y. The vertex must be the middle letter.

X

1

1

360

m1 = 50°

Y

Z

Course 3

Points, Lines, Planes, and Angles

G

H

J

F

K

Course 3

The measures of angles that fit together to form a straight line, such as FKG, GKH, and HKJ, add to 180°.

Points, Lines, Planes, and Angles

P

N

R

Q

M

Course 3

The measures of angles that fit together to form a complete circle, such as MRN, NRP, PRQ, and QRM, add to 360°.

Points, Lines, Planes, and Angles

Course 3

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°.

Points, Lines, Planes, and Angles

Reading Math

A right angle can be labeled with a small box at the vertex.

Course 3

Points, Lines, Planes, and Angles

Course 3

Additional Example 2: Classifying Angles

A. Name a right angle in the figure.

TQS

B. Name two acute angles in the figure.

TQP, RQS

Points, Lines, Planes, and Angles

Course 3

Additional Example 2: Classifying Angles

C. Name two obtuse angles in the figure.

SQP, RQT

Points, Lines, Planes, and Angles

Course 3

Additional Example 2: Classifying Angles

D. Name a pair of complementary angles.

mTQP + mRQS = 47° + 43° = 90°

TQP, RQS

Points, Lines, Planes, and Angles

Course 3

Additional Example 2: Classifying Angles

E. Name two pairs of supplementary angles.

TQP, RQT

mTQP + mRQT = 47° + 133° = 180°

mSQP + mSQR = 137° + 43° = 180°

SQP, SQR

Points, Lines, Planes, and Angles

C

B

90°

A

D

75°

15°

E

Course 3

Check It Out: Example 2

A. Name a right angle in the figure.

BEC

Points, Lines, Planes, and Angles

C

B

90°

A

D

75°

15°

E

Course 3

Check It Out: Example 2

B. Name two acute angles in the figure.

AEB, CED

C. Name two obtuse angles in the figure.

BED, AEC

Points, Lines, Planes, and Angles

C

B

90°

A

D

75°

15°

E

Course 3

Check It Out: Example 2

D. Name a pair of complementary angles.

mAEB + mCED = 15° + 75° = 90°

AEB, CED

Points, Lines, Planes, and Angles

C

B

90°

A

D

75°

15°

E

Course 3

Check It Out: Example 2

E. Name two pairs of supplementary angles.

mAEB + mBED = 15° + 165° = 180°

AEB, BED

mCED + mAEC = 75° + 105° = 180°

CED, AEC

Points, Lines, Planes, and Angles

Course 3

- 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.

Points, Lines, Planes, and Angles

~

So m1 = m3 or m1 = m3.

Course 3

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.

If m1 = 37°, find m3.

The measures of 1 and 2 are supplementary.

m2 = 180° – 37° = 143°

The measures of 2 and 3 are supplementary.

m3 = 180° – 143° = 37°

Points, Lines, Planes, and Angles

So m4 = m2 or m4 m2.

Course 3

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.

If m4 = y°, find m2.

m3 = 180° – y°

m2 = 180° – (180° – y°)

= 180° – 180° + y°

Distributive Property m2 = m4

= y°

Points, Lines, Planes, and Angles

So m1 = m3 or m1 m3.

Course 3

Check It Out: Example 3A

In the figure, 1 and 3 are vertical angles, and 2 and 4 are vertical angles.

2

3

If m1 = 42°, find m3.

1

4

The measures of 1 and 2 are supplementary.

m2 = 180° – 42° = 138°

The measures of 2 and 3 are supplementary.

m3 = 180° – 138° = 42°

Points, Lines, Planes, and Angles

So m4 = m2 or m4 m2.

Course 3

Check It Out: Example 3B

In the figure, 1 and 3 are vertical angles, and 2 and 4 are vertical angles.

2

3

If m4 = x°, find m2.

1

4

m3 = 180° – x°

m2 = 180° – (180° – x°)

= 180° –180° + x°

Distributive Property m2 = m4

= x°

Points, Lines, Planes, and Angles

Possible answer: AD and BE

Course 3

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 m1 = 47°, then find m3.

47°

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