5.3

1. 5.3 Use Angle Bisectors of Triangles B D A C Theorem 5.5: Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the two _______ of the angle.

2. 5.3 Use Angle Bisectors of Triangles B D A C Theorem 5.6: Converse of the Angle Bisector Theorem If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the _________ of the angle.

3. 5.3 Use Angle Bisectors of Triangles Because EC ____, ED _____, and EC = ED = 21, BE bisects CBD by the ____________________________ _________. E 21 C 21 D 31o B Use the Angle Bisector Theorem Example 1 Find the measure of CBE. Solution Converse of the Angle Bisector Theorem

4. 5.3 Use Angle Bisectors of Triangles L R The congruent angles tell you that the spider is on the _________ of LFR. By the _________________________, the spider is equidistant from FL and FR. F Solve a real-world problem Example 2 Web A spider’s position on its web relative to an approaching fly and the opposite sides of the web form congruent angles, as shown. Will the spider have to move farther to reach a fly toward the right edge or the left edge? Solution bisector Angle Bisector Theorem same distance So, the spider must move the ____________ to reach each edge.

5. 5.3 Use Angle Bisectors of Triangles For what value of x does P lie on the bisector of J? From the Converse of the Angle Bisector Theorem, you know that P lies on the bisector of J if P is equidistant from the sides of J, so then _____ = _____. x + 1 K P 2x – 5 J L Point P lies on the bisector of J when x = ____. Use algebra to solve a problem Example 3 Solution PL PK Set segment lengths equal. _____ = _____ ______ = _______ Substitute expressions for segment lengths. ______ = _______ Solve for x.

6. 5.3 Use Angle Bisectors of Triangles B A C Theorem 5.7: Concurrency of Angle Bisector of a Theorem The angle bisector of a triangle intersect at a point that is equidistant from the sides of the triangle. D E P F

7. 5.3 Use Angle Bisectors of Triangles H 15 12 In the diagram, L is the incenter of FHJ. Find LK. G I L J By the Concurrency of Angle Bisectors of a Triangle Theorem, the incenter L is __________ from the sides of FHJ. K So to find LK, you can find ___ in LHI. F Use the concurrency of angle bisectors Example 4 equidistant LI Use the Pythagorean Theorem. Pythagorean Theorem ___ = ________ Substitute known values. ___ = ________ ___ = ____ Simplify. ___ = ____ Solve. LI Because ____ = LK, LK = _____.

8. 5.3 Use Angle Bisectors of Triangles xo 25o Checkpoint. In Exercise 1 and 2, find the value x. 1. Because the segments opposite the angles are perpendicular and congruent, by the Converse of the Angle Bisector Theorem, the ray bisects the angle. So, the angles are congruent, and

9. 5.3 Use Angle Bisectors of Triangles 2. 7x + 3 8x Checkpoint. In Exercise 1 and 2, find the value x. By the Angle Bisector Theorem, the two segments are congruent.

10. 5.3 Use Angle Bisectors of Triangles • Do you have enough information to conclude that AC bisects DAB? B A C D Checkpoint. In Exercise 1 and 2, find the value x. No,

11. 5.3 Use Angle Bisectors of Triangles H G I L J K F Checkpoint. In Exercise 1 and 2, find the value x. • In example 4, suppose you are not given HL or HI, but you are given that JL = 25 and JI = 20. Find LK. 15 12 20 25

12. 5.3 Use Angle Bisectors of Triangles Pg. 289, 5.3 #1-12