standard g.g.13. apply properties of angles, parallel lines, arcs, radii, chords, tangents, and...

Standard

G.G.13. Apply properties of angles, parallel lines, arcs, radii, chords, tangents, and secants to solve problems.

1.4: Measure and Classify Angles1.4: Measure and Classify Angles1.5: Describe Angle Pair Relationships1.5: Describe Angle Pair Relationships

Objectives: By the end of this lesson I will be able to :

1.To define, classify, draw, name, and measure various angles

2.To use the Protractor and Angle Addition Postulates

3.To use special angle relationships to find angle measures

AngleAngle

• An angleangle consists of two different rays (sidessides) that share a common endpoint (vertexvertex).– Angle ABC, ABC,

or B

B

A

C

Sides

Vertex

A “Rabbit Ear” antenna is a physical model of an angle

AngleAngle

• An angleangle consists of two different rays (sidessides) that share a common endpoint (vertexvertex).– Angle ABC, ABC,

or B

Example 1Example 1

How many angles can be seen in the diagram?

Name all the angles.

XY

W

Z

3

<WXY

<YXZ

<WXZ

How Big is an Angle?How Big is an Angle?

Is the angle between the two hands of the wristwatch smaller than the angle between the hands of the large clock?– Both clocks read 9:36

Click me to learn more about measuring angles

Measure of an AngleMeasure of an Angle

The measure of an anglemeasure of an angle is the smallest amount of rotation about the vertex from one side to the other, measured in degrees.

• Can be any value between 0 and 180

• Measured with a protractor

Classifying AnglesClassifying Angles

How To Use a ProtractorHow To Use a Protractor

The measure of this angle is written:

34ABCm

Example 2Example 2

Use the diagram to fine the measure of the indicated angle. Then classify the angle.

1.KHJ

2.GHK

3.GHJ

4.GHL

= 55

= 125

= 180

= 90

Example 3Example 3

. Use your protractor to measure the angles shown for exercises 3-5.

Example 4Example 4

What is the measure of DOZ?

40

25

O

G

D

Z

Example 4Example 4

You basically used the Angle Addition Angle Addition PostulatePostulate to get the measure of the angle, where mDOG + mGOZ = mDOZ.

40

25

O

G

D

Z

Angle Addition PostulateAngle Addition Postulate

If P is in the interior of RST, then mRST = mRSP + mPST.

Example 5Example 5

Given that mLKN = 145°, find mLKM and mMKN.

2x+10

4x-3

K N

M

L

Congruent AnglesCongruent Angles

• Two angles are congruent anglescongruent angles if they have the same measure.

Add the appropriatemarkings to your picture.

Angle BisectorAngle Bisector

An angle bisectorangle bisector is a ray that divides an angle into two congruent angles.

Example 6Example 6

In the diagram, YW bisects XYZ, and mXYW = 18°. Find mXYZ.

X

Y

Z

W

Angle Pair InvestigationAngle Pair Investigation

In this Investigation, you will be shown examples and non-examples of various angle pairs. Use the pictures to come up with a definition of each angle pair.

Complementary AnglesComplementary Angles

Supplementary AnglesSupplementary Angles

C Comes Before S…C Comes Before S…

9043

9021

mm

mm

18087

18065

mm

mm

Linear Pairs of AnglesLinear Pairs of Angles

Linear Pairs of AnglesLinear Pairs of Angles

• Two adjacent angles form a linear pair linear pair if their noncommon sides are opposite rays.

• The angles in a linear pair are supplementarysupplementary.

Vertical AnglesVertical Angles

Vertical AnglesVertical Angles

• Two nonadjacent angles are vertical vertical anglesangles if their sides form two pairs of opposite rays.

• Vertical angles are formed by two intersecting lines.

Example 7 Example 7

1. Given that 1 is a complement of 2 and m1 = 68°, find m2.

2. Given that 3 is a complement of 4 and m3 = 56°, find m4.

Example 8Example 8

Identify all of the linear pairs of angles and all of the vertical angles in the figure.

Example 9Example 9

Two angles form a linear pair. The measure of one angle is 5 times the measure of the other angle. Find the measure of each angle.