1.3 Angles and Their Measures

Warm Up

1. Draw opposite rays and .

Solve each equation.

3. 2x + 3 + x – 4 + 3x – 5 = 180

2. Draw and , where A, B, and C are noncollinear.

4. 5x + 2 = 8x – 10

Name and classify angles.

Measure and construct angles and angle bisectors.

Objectives

1.3 Angles and Their Measures

angle is a figure formed by two ______, or sides, with a common _____________ called the vertex (plural: vertices).

R,

Angle Name

SRT, TRS, or 1

You cannot name an ______ just by its ______ if the point is the vertex of more than ____ _______. In this case, you must use all _____ _______ to name the angle, and the _______ point is always the vertex.

angle vertexone

three pointsmiddle

angle

raysendpoint

Write the different ways you can name the angles in the diagram.

Example 1

The measure of an angle is usually given in degrees. Since there are 360° in a circle, one degree is of a circle. When you use a protractor to measure angles, you are applying the following postulate.

Classify each angle as acute, right, or obtuse.

A. XTS

B. WTU

C. XTU

Example 2

Example 3Find the measure of each angle.

Use a protractor to draw an angle with a measure of 165°.

Example 4

Find the measure of each angle. Then classify each as acute, right, or obtuse.

Example 5

C. WXV

B. ZXW

A. YXW

Example 6

Find the measure of each angle. Then classify each as acute, right, or obtuse.

C. COD

B. BOCA. AOB

D. DOB

Congruent angles are angles that have the same ___________.Arc marks are used to show that the two angles are congruent.

measure

In the diagram, mABC = mDEF, so ABC DEF.

mDEG = 115°, and mDEF = 48°. Find mFEG

Example 7

Example 8

K is in the interior of LMN, mLMK =52°, and mKMN = 12°. Draw the picture. Find mLMN.

An angle bisector is a _____ that divides an angle into ____ _________ ______.

raytwo congruent angles

bisects LJM; thus LJK KJM.

bisects JKL, mJKM = (4x + 6)°, and mMKL = (7x – 12)°. Find mJKM.

Example 9

Example 10

bisects LJM, mLJK = (-10x + 3)°, andmKJM = (–x + 21)°. Find mLJM.

Example 11

bisects PQR, mPQS = (5y – 1)°, andmPQR = (8y + 12)°. Find mPQS.

bisects ABC, mABD = (y + 10) °, and mDBC = (y + 4)°.

Find mABC.

Example 12