1.4: Measure and Classify Angles Objectives: 1.To define, classify, draw, name, and measure various...

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

In your notes, 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, CBA or B

• The vertex is the key

B

A

C

Sides

Vertex

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

AngleAngle

• Each angles creates 2 regions: INTERIOR INTERIOR and EXTERIOR. EXTERIOR.

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

http://www.mathopenref.com/degrees.html

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

Click the button to see how to measure an angle.

http://www.mathopenref.com/constmeasureangle.html

Example 3Example 3

What is the measure of DOZ?

40

25

O

G

D

Z

650

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

LSet up an equation and solve

X = 23<LKM = 56<MKN = 89CHECK: 56 + 89 = 145

Congruent AnglesCongruent Angles

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

Add the appropriatemarkings to your picture.


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

compass and a

straightedge?


1. Draw angle A.


2. Draw a ray with endpoint A’.


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

B and C.


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.


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


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’.


7. Draw ray A’C’.


Click on the button to watch a video of the construction.

http://www.mathopenref.com/constcopyangle.html

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.


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


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.


4. Label the intersection D.


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

Angle Bisector: VideoAngle Bisector: Video

Click on the button to watch a video of the construction.

http://www.mathopenref.com/constbisectangle.html

Example 5Example 5

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

X

Y

Z

W

360

Example 6Example 6

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

Set up and equation and solve.

X = 3Both angles are 38

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. = ? 180 27

2.Convert rad into degrees.4

3

27 π180

3 20

= 3/4π180 ?

? = 135

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