5.3Use Angle Bisectors of Triangles 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. A C B D

5.3Use Angle Bisectors of Triangles 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. A C B D

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

5.3Use Angle Bisectors of Triangles Example 2 Solve a real-world problem 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? F L R Solution The congruent angles tell you that the spider is on the _________ of LFR. bisector By the _________________________, the spider is equidistant from FL and FR. Angle Bisector Theorem So, the spider must move the ____________ to reach each edge. same distance

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 _____ = _____. 5.3Use Angle Bisectors of Triangles Example 3 Use algebra to solve a problem Solution PK For what value of x does P lie on the bisector of J? K J P L x + 1 2x – 5 PL Set segment lengths equal. _____ = _____ ______ = _______ Substitute expressions for segment lengths. Solve for x. ______ = _______ Point P lies on the bisector of J when x = ____.

5.3Use Angle Bisectors of Triangles 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. A B C P E D F

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

5.3Use Angle Bisectors of Triangles Checkpoint. In Exercise 1 and 2, find the value x. 1. xoxo 25 o 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

5.3Use Angle Bisectors of Triangles Checkpoint. In Exercise 1 and 2, find the value x. By the Angle Bisector Theorem, the two segments are congruent. 2. 7x + 3 8x8x

5.3Use Angle Bisectors of Triangles Checkpoint. In Exercise 1 and 2, find the value x. 3.Do you have enough information to conclude that AC bisects DAB? A C D B No,

5.3Use Angle Bisectors of Triangles Checkpoint. In Exercise 1 and 2, find the value x. 3.In example 4, suppose you are not given HL or HI, but you are given that JL = 25 and JI = 20. Find LK. I H K G F J L

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