1.4 Angle Measures

Objectives: How to label, measure, and classify

angles Identifying and using congruent

angles Creating and utilizing an angle

bisector

Rays To create angles we must first define

a ray. A ray is part of a line which has one endpoint and extends infinitely in one direction. Rays are named stating the endpoint first and then any other point on the ray.

Labeling Rays We could label

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

A

D

C

B

More about Rays If you choose a point on a line, that

point determines exactly two rays called opposite rays.

Q RP

QP and QR are opposite rays.

Angles and Their Parts An angle is formed by two

noncollinear 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

A1

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

A1

F DE

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.

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.

BAC YXZA

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. 34 – 35, #12 – 37, 42 - 43

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