5.3 Using Angle Bisectors of Triangles

1. 5.3 Using Angle Bisectors of Triangles

2. Vocabulary/Theorems • Angle bisector: ray that divides angle into 2 congruent angles • Point of concurrency: point of intersection of segments, lines, or rays • Incenter: point of concurrency of angle bisectors of a triangle • Angle Bisector Theorem: If a point is on the bisector of an angle, then it is equidistant from the 2 sides. (distance from point to a line is a perp. path)

3. Vocabulary/Theorems • Converse of Angle Bisector Theorem: • Angle bisectors intersect at a point that is equidistant from the sides of a triangle. (Incenter is equidistant from sides)

4. 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.

5. With a partner, do #1-3 on p. 273

6. B B P A P A C C ANSWER ANSWER ANSWER 11 5 15 for Examples 1, 2, and 3 GUIDED PRACTICE In Exercises 1–3, find the value of x. 1. 2. 3. P B C A

7. 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

9. In the diagram, Nis the incenter of ABC. Find ND. By the Concurrency of Angle Bisectors of a Triangle Theorem, the incenter Nis equidistant from the sides of ABC. So, to find ND, you can find NFin NAF. Use the Pythagorean Theorem stated on page 18. EXAMPLE 4 Use the concurrency of angle bisectors SOLUTION

10. 2 2 2 c = a + b 400 = 2 NF + 256 2 144 = NF 12 = NF 2 2 2 20 = NF + 16 EXAMPLE 4 Use the concurrency of angle bisectors Pythagorean Theorem Substitute known values. Multiply. Subtract 256 from each side. Take the positive square root of each side. Because NF = ND, ND = 12.

11. Inscribed Circle • The incenter is the center of the inscribed circle.