1.4 Measure and Classify Angles

Objectives:

How to label, measure, and classify angles

Identifying and using congruent angles

Creating and utilizing an angle bisector

Angles and Their Parts

An angle is formed by two non-collinear rays that have a common endpoint. The rays are called sides and the common endpoint is the vertex. B

CA

Side AB

Vertex A

Side AC

Labeling Angles

We label angles any of the following ways:

BAC, CAB, A, or 1

C

B

A

1

More about Angles

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

Name the sides of 5.

Answer: and or are the sides of 5.

Example 1b:

Write another name for 6.

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

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:

Measuring Angles

To measure an angle we use a protractor. Place the center of the protractor on the vertex and one side of the angle on either side of the 0° line of the protractor. The protractor will have two scales running from 0° to 180° in opposite directions. Read the measure of the angle by viewing the alignment of the other side of the angle with the proper scale.

Postulates

Postulate 3 (Protractor Postulate)All angles have measures between 0° and 180°.

Postulate 4 (Angle Addition Postulate) R is in the interior of PQS iff mPQR + mRQS = mPQS.

S

P

Q

R

Classifying Angles

There are four types of angles.

Acute angles measure < 90°. Right angles measure 90°.

Obtuse angles measure > 90° but < 180°.

Straight 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:

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

Use a protractor to find that .

Answer: >is an obtuse angle.

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:

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:

Congruent Angles

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

BAC YXZ

A

YB

X ZC

More about Congruent Angles

A ray that divides an angle into two congruent angles is called an angle bisector. If AD bisects BAC then 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:

Given

Definition of congruent angles

Substitution

Add 10 to each side.

Subtract 2x from each side.

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:

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:

Assignment:

Geometry: Pg. 28 – 32

#3 – 27, 33 – 38, 40 - 42