5 3 Use Angle Bisectors of Triangles Theorem
5. 3 Use 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 B D C
5. 3 Use 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 A the _____ of the angle. B D C
5. 3 Use Angle Bisectors of Triangles Example 1 Use the Angle Bisector Theorem Find the measure of CBE. Solution Because EC ____, ED _____, and EC = ED = 21, BE bisects CBD by Converse of the Angle Bisector the ______________ Theorem _____. 21 E C 21 31 B o D
5. 3 Use 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? L Solution F The congruent angles tell you that the spider is bisector of LFR. on the _____ Angle Bisector Theorem By the _____________, the spider is equidistant from FL and FR. same distance to So, the spider must move the ______ reach edge. R
5. 3 Use Angle Bisectors of Triangles Example 3 Use algebra to solve a problem For what value of x does P lie on the bisector of J? Solution From the Converse of the Angle Bisector x+1 K Theorem, you know that P lies on the bisector of J if P is equidistant from PL the sides of J, so then _____ PK = _____. Set segment J _____ = _____ lengths equal. L ______ = _______ Substitute expressions for segment lengths. ______ = _______ Solve for x. Point P lies on the bisector of J when x = ____. P 2 x – 5
5. 3 Use 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 D E P F C
5. 3 Use Angle Bisectors of Triangles Example 4 Use the concurrency of angle bisectors In the diagram, L is the incenter of FHJ. Find LK. By the Concurrency of Angle Bisectors of a Triangle Theorem, the incenter L equidistant from the sides of FHJ. is _____ So to find LK, you can find ___ LI in LHI. Use the Pythagorean Theorem. ___ = ________ F ___ = ____ Because ____ LI = LK, LK = _____. 15 H 12 G I L J K Pythagorean Theorem Substitute known values. Simplify. Solve.
5. 3 Use Angle Bisectors of Triangles Checkpoint. In Exercise 1 and 2, find the value x. 1. xo 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. 3 Use Angle Bisectors of Triangles Checkpoint. In Exercise 1 and 2, find the value x. 2. 7 x + 3 8 x By the Angle Bisector Theorem, the two segments are congruent.
5. 3 Use 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? B A C D No,
5. 3 Use 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. 15 H 12 G I L K F 20 J 25
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