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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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