5-1 Points, Lines, Planes, and Angles

Pre-Algebra

Warm Up

Problem of the Day

Lesson Presentation

WARM-UPMrs. 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?1

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Warm UpSolve.

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

Learn to classify and name figures.

Plane Geometry Vs Solid GeometryCan you explane the difference?

explain

YOU NEED TO KNOW THESE WORDS…see 5-1 in book

point line plane

segment ray angle

rightiangle acuteiiangle

obtuseiiangle complementaryiiangles

supplementaryiiangles

vertical angles

congruent

If you didn’t have time to write them all down, don’t bang you head! Just look in your book and follow along with me! You should be doing this anyhow!...lesson 5-1

Points, lines, and planes are the building blocks of

geometry. Segments, rays, and angles are defined in

terms of these basic figures. Let’s look more closely at these things!

A point names a location.

• A Point A

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

line l, or BCB

Cl

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

plane P, or plane DEF

DE

F

PP

G

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

GH

K

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

KJ

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

A. Name 4 points in the figure.

B. Name a line in the figure.

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

Any 2 points on a line can be used.KL or JK

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

C. Name a plane in the figure.

Plane , plane JKL 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.KJ, KL, JK, LK

JK, KL, LM, JM

Try This: Example 1A & 1B

A. Name 4 points in the figure.

B. Name a line in the figure.

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

A B

CD

DA or BC Any 2 points on a line can be used.

C. Name a plane in the figure.

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

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

A B

CD

Try This: Example 1C

D. Name four segments in the figure

E. Name four rays in the figureDA, AD, BC, CB

AB, BC, CD, DA

A B

CD

Try This: Example 1D & 1E

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.

1360

X

Y Z1 m1 = 50°

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

F K J

G H

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

P

R QM

N

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°.

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

Reading Math

Additional Example 2A & 2B: Classifying Angles

A. Name a right angle in the figure.

B. Name two acute angles in the figure.

TQS

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.

TQP, RQS mTQP + mRQS = 47° + 43° = 90°

Additional Example 2E: Classifying Angles

E. Name two pairs of supplementary angles.

TQP, RQT

SQP, RQS

mTQP + mRQT = 47° + 133° = 180°

mSQP + mRQS = 137° + 43° = 180°

Try This: Example 2A

A. Name a right angle in the figure.

BEC

E

D

CB

A 90° 75°15°

C. Name two obtuse angles in the figure.

BED, AEC

B. Name two acute angles in the figure.

AEB, CED

E

D

CB

A 90° 75°15°

Try This: Example 2B & 2C

D. Name a pair of complementary angles.

AEB, CED

E

D

CB

A 90° 75°15°

Try This: Example 2D

mAEB + mCED = 15° + 75° = 90°

E. Name two pairs of supplementary angles.

AEB, BED

CED, AEC

E

D

CB

A 90° 75°15°

Try This: Example 2D & 2E

mAEB + mBED = 15° + 165° = 180°

mCED + mAES = 75° + 105° = 180°

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 m1 = 37°, find m3.

The measures of 1 and 2 add to 180° because they are supplementary, so m2 = 180° – 37° = 143°.

The measures of 2 and 3 add to 180° because they are supplementary, so m3 = 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 m4 = y°, find m2.

m3 = 180° – y°

m2 = 180° – (180° – y°)

= 180° – 180° + y°

= y°

Distributive Property m2 = m4

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

A. If m1 = 42°, find m3.

The measures of 1 and 2 add to 180° because they are supplementary, so m2 = 180° – 42° = 138°.

The measures of 2 and 3 add to 180° because they are supplementary, so m3 = 180° – 138° = 42°.

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Try This: Example 3A

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

B. If m4 = x°, find m2.

m3 = 180° – x°

m2 = 180° – (180° – x°)

= 180° –180° + x°

= x°

Distributive Property m2 = m4

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Try This: Example 3B

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.

3. Name a right angle in the figure.

4. Name a pair of complementary angles.

5. If m1 47°, then find m3.

2. Name two lines in the figure.

Possible answer: A, B, and C

Possible answer: AGF

Possible answer: 1 and 2

47°

Possible answer: AD and BE