CK-12 Geometry Concepts

1.3. Congruent Angles and Angle Bisectors http://www.ck12.org

Anangle bisectoris a ray that divides an angle into two congruent angles, each having a measure exactly half of the
original angle. Every angle has exactly one angle bisector.

BDis the angle bisector of^6 ABC

(^6) ABD∼= (^6) DBC
m^6 ABD=

1

2

m^6 ABC

Label equal angles withangle markings, as shown below.

1. Draw an angle on your paper. Make sure one side is horizontal.


2. Place the pointer on the vertex. Draw an arc that intersects both sides.


3. Move the pointer to the arc intersection with the horizontal side. Make a second arc mark on the interior of
   the angle. Repeat on the other side. Make sure they intersect.


4. Connect the arc intersections from #3 with the vertex of the angle.

To see an animation of this construction, view http://www.mathsisfun.com/geometry/construct-anglebisect.html.

Example A

How many angles are in the picture below? Label each one two different ways.

There are three angles with vertexU. It might be easier to see them all if we separate them out.

So, the three angles can be labeled,^6 XUYor^6 Y U X,^6 Y U Zor^6 ZUY, and^6 XU Zor^6 ZU X.

Example B

What is the measure of each angle?

From the picture, we see that the angles are congruent, so the given measures are equal.

( 5 x+ 7 )◦= ( 3 x+ 23 )◦

2 x◦= 16 ◦

x= 8 ◦

To find the measure of^6 ABC, plug inx= 8 ◦to( 5 x+ 7 )◦.

( 5 ( 8 )+ 7 )◦

( 40 + 7 )◦

47 ◦

Becausem^6 ABC=m^6 XY Z,m^6 XY Z= 47 ◦too.