1. 5.3 – Use Angle Bisectors of Triangles
Remember that an angle bisector is a ray that divides an angle into two congruent adjacent angles.
Remember also that the distance from a point to a line is the length of the perpendicular segment from the point to the line.

2. 5.3 – Use Angle Bisectors of Triangles
In the diagram, Ray PS is the bisector of <QPR and the distance from S to Ray PQ is SQ, where Segment SQ is perpendicular to Ray PQ.

3. 5.3 – Use Angle Bisectors of Triangles

4. 5.3 – Use Angle Bisectors of Triangles
Example 1:
Find the measure of <GFJ.

5. 5.3 – Use Angle Bisectors of Triangles
Example 2:
Three spotlights from two congruent angles. Is the actor closer to the spotlighted area on the right or on the left?

6. 5.3 – Use Angle Bisectors of Triangles
Example 3:
For what value of x does P lie on the bisector of <A?

7. 5.3 – Use Angle Bisectors of Triangles
Example 4:
Find the value of x.

8. 5.3 – Use Angle Bisectors of Triangles

9. 5.3 – Use Angle Bisectors of Triangles
The point of concurrency of the three angle bisectors of a triangle is called the incenter of the triangle. The incenter always lies inside the triangle.

10. 5.3 – Use Angle Bisectors of Triangles
Example 5:
In the diagram, N is the incenter of Triangle ABC. Find ND.

11. 5.3 – Use Angle Bisectors of Triangles
Example 6:
In the diagram, G is the incenter of
Triangle RST. Find GW.