EXAMPLE 1 Use the Angle Bisector Theorems SOLUTION Because JG FG and JH FH and JG = JH = 7, FJ bisects GFH by the Converse of the Angle Bisector Theorem.

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EXAMPLE 1 Use the Angle Bisector Theorems SOLUTION Because JG FG and JH FH and JG = JH = 7, FJ bisects GFH by the Converse of the Angle Bisector Theorem. So, m GFJ = m HFJ = 42. o Find the measure of ∠ GFJ.

EXAMPLE 2 Solve a real-world problem SOCCER 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 R or the left goalpost L ?

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

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

GUIDED PRACTICE for Examples 1, 2, and 3 In Exercises 1–3, find the value of x. SOLUTION From the Converse of the Angle Bisector Theorem, you know that P lies on the bisector of A if P is equidistant from the sides of A, so when BP = CP. BP = CP Set segment lengths equal. 15 = x Solve for x. 1. A B C P

GUIDED PRACTICE for Examples 1, 2, and 3 In Exercises 1–3, find the value of x. BAP = CAP 3x + 5 = 4x – 6 11 = x Set angle equal. Substitute expressions for segment lengths. Solve for x. A B C P 2. SOLUTION From the Converse of the Angle Bisector Theorem, you know that P lies on the bisector of A if P is equidistant from the sides of A, so when BAP = CAP

GUIDED PRACTICE for Examples 1, 2, and 3 In Exercises 1–3, find the value of x. 3. SOLUTION From the Converse of the Angle Bisector Theorem, you know that P lies on the bisector of A if P is equidistant from the sides of A, so when BP = CP. BP = CP Set segment lengths equal. 5x = 6x –5 Substitute expressions for segment lengths. 5 = x Solve for x. A BC P

GUIDED PRACTICE for Examples 1, 2, and 3 4. Do you have enough information to conclude that QS bisects PQR ? Explain. SOLUTION No; you need to establish that SR QR and SP QP. In Exercises 1–3, find the value of x.