1.4 Angle Measures1.4 Angle Measures

Objectives:Objectives:

How to label, measure, and classify How to label, measure, and classify anglesangles

Identifying and using congruent Identifying and using congruent anglesangles

Creating and utilizing an angle Creating and utilizing an angle bisectorbisector

RaysRays

A A ray ray is part of a line which has one is part of a line which has one

endpoint and extends infinitely in endpoint and extends infinitely in one direction. one direction.

named stating the endpoint first and named stating the endpoint first and then any other point on the ray. then any other point on the ray.

Labeling RaysLabeling Rays

We could label We could label this ray as AB, this ray as AB, AC, or AD but not AC, or AD but not CA.CA.

A

D

C

B

More about RaysMore about Rays

If you choose a point on a line, that If you choose a point on a line, that point determines exactly two rays point determines exactly two rays called called opposite raysopposite rays..

Q RP

QP and QR are opposite rays.

Angles and Their PartsAngles and Their Parts

An An angleangle is formed by two is formed by two noncollinear rays that have a noncollinear rays that have a common endpoint. common endpoint. – the rays are called the rays are called sidessides – common endpoint is the common endpoint is the vertexvertex. . B

CA

Side AB

Vertex A

Side AC

Labeling AnglesLabeling Angles

We label angles any of the following We label angles any of the following ways:ways:

BAC, CAB, A, or 1BAC, CAB, A, or 1

C

B

A

1

More about AnglesMore about Angles

An angle divides a plane into three An angle divides a plane into three distinct parts. Points A, B, and C lie distinct parts. Points A, B, and C lie on the angle. Points D and E line in on the angle. Points D and E line in the the interiorinterior of the angle. Points F of the angle. Points F and G lie in the and G lie in the exteriorexterior of the of the angle.angle.

C

B

A

1F D

E

G

Name all angles that have B as a vertex.

Answer: 5, 6, 7, and ABG

Example 1a:Example 1a:

Name the sides of 5.

Answer: and or are the sides of 5.

Example 1b:Example 1b:

Write another name for 6.

Answer: EBD, FBD, DBF, and DBE are other names for 6.

Example 1c:Example 1c:

a. Name all angles that have X as a vertex.

b. Name the sides of 3.

c. Write another name for 3.

Answer: 1, 2, 3, and RXB or RXN

Answer: AXB, AXN, NXA, BXA

Answer:

Your Turn:Your Turn:

Measuring AnglesMeasuring Angles

To measure an angle we use a To measure an angle we use a protractorprotractor. . – Place the center of the protractor on the Place the center of the protractor on the

vertex and one side of the angle on either vertex and one side of the angle on either side of the 0side of the 0° line of the protractor. ° line of the protractor.

– The protractor will have two scales running The protractor will have two scales running from 0° to 180° in opposite directions. from 0° to 180° in opposite directions.

– Read the measure of the angle by viewing Read the measure of the angle by viewing the alignment of the other side of the angle the alignment of the other side of the angle with the proper scale. with the proper scale.

Classifying AnglesClassifying Angles

There are four types of angles.There are four types of angles.

AcuteAcute angles measure < angles measure < 9090°.°.

RightRight angles measure 90 angles measure 90°.°.

ObtuseObtuse angles measure > angles measure > 9090° but < ° but < 180°.180°.

StraightStraight angles measure 180°.angles measure 180°.

Measure TYV and classify it as right, acute, or obtuse.

TYV is marked with a right angle symbol, so measuring is not necessary.

Answer: is a right angle.

Example 2a:Example 2a:

Measure WYT and classify it as right, acute, or obtuse.

Use a protractor to find that .

Answer: >is an obtuse angle.

Example 2b:Example 2b:

Measure TYU and classify it as right, acute, or obtuse.

Use a protractor to find that m .

Answer: is an acute angle.

Example 2c:Example 2c:

Measure each angle named and classify it as right, acute, or obtuse.

a. CZD

b. CZE

c. DZX

Answer: 150, obtuse

Answer: 90, right

Answer: 30, acute

Your Turn:Your Turn:

Congruent AnglesCongruent Angles

Just as segments that have equal Just as segments that have equal measures are measures are congruentcongruent,, angles that angles that have the same measures are congruent. have the same measures are congruent. To label angles congruent we use tic To label angles congruent we use tic marks just like we used for segments.marks just like we used for segments.

BAC BAC YXZ YXZ

A

YB

X ZC

More about Congruent More about Congruent AnglesAngles

A ray that divides an angle into two A ray that divides an angle into two congruent angles is called an congruent angles is called an angle angle bisectorbisector. If AD bisects BAC then . If AD bisects BAC then BAD is congruent to CAD. BAD is congruent to CAD.

C

B

A

D

INTERIOR DESIGN Wall stickers of standard shapes are often used to provide a stimulating environment for a young child’s room. A five-pointed star sticker is shown with vertices labeled. Find mGBH and mHCI if GBH HCI, mGBH 2x + 5, and mHCI 3x – 10.

Example 3:Example 3:

Given

Definition of congruent angles

Substitution

Add 10 to each side.

Subtract 2x from each side.

Example 3:Example 3:

Given

Simplify.

Use the value of x to find the measure of one angle.

Since .

or 35

Answer: Both measure 35.

Example 3:Example 3:

SIGNS A railroad crossing sign forms congruent angles. In the figure, WVX ZVY. If mWVX 7a + 13 and mZVY 10a – 20, find the actual measurements of WVX and ZVY.

Answer:

Your Turn:Your Turn:

Assignment:Assignment:

Geometry:Geometry: Pg. 34 – 35, #12 - 37Pg. 34 – 35, #12 - 37

Pre-AP Geometry:Pre-AP Geometry:Pg. 34 – 35, #12 – 39, 45 - 48Pg. 34 – 35, #12 – 39, 45 - 48