EXAMPLE 1

1. SOLUTION Because JG FGand JH FHand JG =JH = 7, FJ bisects GFH by the Converse of the Angle Bisector Theorem. So,mGFJ =mHFJ = 42°. EXAMPLE 1 Use the Angle Bisector Theorems Find the measure of GFJ.

2. A soccer goalie’s position relative to the ball and goalposts forms congruent angles, as shown. Will the goalie have to move farther to block a shot toward the right goalpost Ror the left goalpost L? EXAMPLE 2 Solve a real-world problem SOCCER

3. The congruent angles tell you that the goalie is on the bisector of LBR. By the Angle Bisector Theorem, the goalie is equidistant from BRand BL. So, the goalie must move the same distance to block either shot. EXAMPLE 2 Solve a real-world problem SOLUTION

4. For what value of xdoes Plie on the bisector of A? From the Converse of the Angle Bisector Theorem, you know that P lies on the bisector of Aif Pis equidistant from the sides of A, so when BP =CP. CP BP = 2x –1 x + 3 = 4 = x Point Plies on the bisector of Awhen x = 4. EXAMPLE 3 Use algebra to solve a problem ALGEBRA SOLUTION Set segment lengths equal. Substitute expressions for segment lengths. Solve for x.

5. B P A C ANSWER 15 for Examples 1, 2, and 3 GUIDED PRACTICE In Exercises 1–3, find the value of x. 1.

6. B P A C ANSWER 11 for Examples 1, 2, and 3 GUIDED PRACTICE In Exercises 1–3, find the value of x. 2.

7. ANSWER 5 for Examples 1, 2, and 3 GUIDED PRACTICE In Exercises 1–3, find the value of x. 3. P B C A

8. Do you have enough information to conclude that QSbisects PQR? Explain. 4. ANSWER No; you need to establish thatSR QRand SP QP. for Examples 1, 2, and 3 GUIDED PRACTICE