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. To Construct Angle Bisectors • Place point of compass on the angle vertex • Draw an arc through both adjacent sides of the triangle • Move the point of the compass to one of the intersection points of the arc and the side • Open the compass ½ the distance between the 2 sides and create an arc above the current one • Move the compass to the other side and repeat making the 2 arcs intersect • Using a straightedge, connect the vertex with this new arc intersection

5. To Construct Angle Bisectors If repeated with the 3 angle, the bisectors would meet at the point of concurrency, the incenter. The incenter is equidistant to each side of the triangle.

6. 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.

7. 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? 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

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

9. 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 SOLUTION Set segment lengths equal. Substitute expressions for segment lengths. Solve for x.

10. B P A 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. B P A C P 3. B C A

11. 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

12. Do #5 on p. 273

13. 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

14. 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.