Definition of Angle Bisector

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

Constructing an Angle Bisector (Interactively!)

For any angle, follow these steps. (Use the interactive math tool below for an interactive construction.)

• Step 1. Draw a circle with center at the vertex of the angle.
• Step 2. Mark the points of intersection P and Q of this circle with the two sides of the angle.
• Step 3. Draw two circles with the same radius with centers at the points P and Q. The radius must be long enough for the circles to intersect.
• Step 4. Mark a point of intersection D of the two circles that lies in the interior of the angle.
• Step 5. Draw a ray from the vertex of the angle through point D. This ray is the angle bisector.

You can view these steps in the following applet. Drag point B to change the size of the angle. Use the Measure button to verify that the angle bisector has been constructed.

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A Closer Look at the Construction of an Angle Bisector—Why Does It Work?

In the following diagram, drag point B to change the size of the angle. Look for relationships and answer the questions below.

• How are segments OP and OQ related? Explain.
• How are segments DP and DQ related?
• What can you say about triangles ODP and ODQ? Why?
• What does this imply about angles POD and QOD?

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Answer

Since segments OP and OQ are radii of the same circle, they are congruent. Segments DP and DQ are congruent because they are radii of two congruent circles. Triangles ODP and ODQ have two pairs of congruent sides, pair OP and OQ and pair DP and DQ, and they share their third side, OD. This means that triangles ODP and ODQ are congruent triangles and so corresponding angles DOP and DOQ have equal measure. Thus, OD must be the angle bisector of angle AOB.

What Are Some Properties and Applications of Angle Bisectors?

• Explore the relationship among the three angle bisectors of a triangle
  
• See how to use angle bisectors to construct the incircle of a triangle

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References and Credits