1.4: Measure and Classify Angles

1.4: Measure and Classify Angles Objectives: • To define, classify, draw, name, and measure various angles • To use the Protractor and Angle Addition Postulates • To construct congruent angles and angle bisectors with compass and straightedge • To convert angle measurement between degrees and radians

Vocabulary 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.

Angle • An angle consists of two different rays (sides) that share a common endpoint (vertex). • Angle ABC, ABC, or B Vertex Sides A “Rabbit Ear” antenna is a physical model of an angle

Angle • An angle consists of two different rays (sides) that share a common endpoint (vertex). • Angle ABC, ABC, or B

Example 1 How many angles can be seen in the diagram? Name all the angles.

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 Angle The measure 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 Angles Surely you are familiar with all of my angular friends by now.

The measure of this angle is written: How To Use a Protractor

Example 2 Complete your Protractor Practice worksheet. • Write your answer in the form • Draw your angles on the back and label them something!

Example 3 What is the measure of DOZ?

Example 3 You basically used the Angle Addition Postulate to get the measure of the angle, where mDOG + mGOZ = mDOZ.

Angle Addition Postulate If P is in the interior of RST, then mRST= mRSP + mPST.

Example 4 Given that mLKN = 145°, find mLKM and mMKN.

Congruent Angles • Two angles are congruent angles if they have the same measure. Add the appropriate markings to your picture.

Congruent 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 Angles • Draw angle A.

Congruent Angles • Draw a ray with endpoint A’.

Congruent Angles • Put point of compass on A and draw an arc that intersects both sides of the angle. Label these points B and C.

Congruent Angles • 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’ where the arc intersects the ray.

Congruent Angles • Put point of compass on B and pencil on C. Make a small arc.

Congruent Angles • 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 Angles • Draw ray A’C’.

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

Bisect an Angle • Draw an acute angle and label the vertex A.

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

Bisect an Angle • 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 Angle • Label the intersection D.

Bisect an Angle • Draw a ray from vertex A through point D.

Example 5 In the diagram, YW bisects XYZ, and mXYW = 18°. Find mXYZ.

Example 6 In the diagram, OE bisects angle LON. Find the value of x and the measure of each angle.

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

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

Example 7 How many radians would be the equivalent to one full revolution around the unit circle? How many radians would equal 180°?

Example 8 Use the conversion factor 180° =  radians to convert the following angle measures. • Convert 27° into radians. • Convert rad into degrees.

1.4: Measure and Classify Angles