1.4: Measure and Classify 1.4: Measure and Classify AnglesAngles

Objectives:1.To define, classify, draw, name, and

measure various angles2.To use the Protractor and Angle Addition

Postulates3.To construct congruent angles and angle

bisectors with compass and straightedge4.To convert angle measurement between

degrees and radians

VocabularyVocabulary

Angle Obtuse Vertex Right Sides Straight Acute Congruent ’s

As a group, define each of these without your book. Draw a picture for each word and leave a bit of space for additions and revisions.

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

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

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

Surely you are familiar with all of my angular friends by now.

How To Use a ProtractorHow To Use a Protractor

The measure of this angle is written:

34ABCm

Example 2Example 2

Complete your Protractor Practice worksheet.

1.Write your answer in the form

2.Draw your angles on the back and label them something!

m ABC

Example 3Example 3

What is the measure of DOZ?

40

25

O

G

D

Z

Example 3Example 3

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 4Example 4

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.

Congruent AnglesCongruent Angles

Draw angle A on your paper. How could you copy that angle to another part of your paper using only a

compass and a

straightedge?

Congruent AnglesCongruent Angles

1. Draw angle A.

Congruent AnglesCongruent Angles

2. Draw a ray with endpoint A’.

Congruent AnglesCongruent Angles

3. Put point of compass on A and draw an arc that intersects both sides of the angle. Label these points

B and C.

Congruent AnglesCongruent Angles

4. Put point of compass on A’ and use the compass setting from Step 3 to draw a similar arc on the ray.

Label point B’ wherethe arc intersects the ray.

Congruent AnglesCongruent Angles

5. Put point of compass on B and pencil on C. Make a small arc.

Congruent AnglesCongruent Angles

6. Put point of compass on B’ and use the compass setting from Step 5 to draw an arc that intersects the

arc from Step 4.

Label the

new point

C’.

Congruent AnglesCongruent Angles

7. Draw ray A’C’.

Angle BisectorAngle Bisector

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

Bisect an AngleBisect an Angle

1. Draw an acute angle and label the vertex A.

Bisect an AngleBisect an Angle

2. Using vertex A as the center, draw an arc intersecting both sides of your angle. Label the intersections B and C.

Bisect an AngleBisect an Angle

3. Using the same compass setting, draw two intersecting arcs in the interior of your angle, one centered at B, the other centered at C.

Bisect an AngleBisect an Angle

4. Label the intersection D.

Bisect an AngleBisect an Angle

5. Draw a ray from vertex A through point D.

Example 5Example 5

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

X

Y

Z

W

Example 6Example 6

In the diagram, OE bisects angle LON. Find the value of x and the measure of each angle.

RadiansRadians

You can also measure an angle in radiansradians. Radians are like the less well-known greasy, nerdy-type who eats lots of pie.

RadiansRadians

One radian One radian is the measure of the angle formed by stretching the radius of a circle around its circumference.

Example 7Example 7

How many radians would be the equivalent to one full revolution around the unit circle?

How many radians would equal 180°?

Example 8Example 8

Use the conversion factor 180° = radians to convert the following angle measures.

1.Convert 27° into radians.

2.Convert rad into degrees.4

3