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5.3 Use Angle Bisectors of a Triangle angle bisector : a ray that divides an angle into two congruent adjacent angles. distance from a point to a line : the length of the perpendicular segment from the point to the line P Thm 5.5 Angle Bisector Theorem If a point is on the bisector of an angle, then it is__________________ from the sides of the angle. B -If AD bisects ∠BAC and DB ⊥ AB and DC ⊥ AC, then D A C Thm 5.6 Converse of the Angle Bisector Theorem If a point is in the interior of an angle is equidistant from the sides and of the angle, then it lies ______________________________ B D A C 1 D Ex 1: Find the measure of ∠BAD. 18 18 B 54º A Ex 2: Find the value of x. State the theorem used in each problem. a. (3x + 5)º (4x 6)º b. 5x 6x - 5 C D C B A Thm 5.7 Concurrency of Angle Bisectors of a Triangle -The ∠ bisectors of a triangle equidistant from the side intersect at a point that is of the triangle. B F P A E Incenter: the point of concurrency of the angle bisectors of a triangle Ex 3: The ∠ bisectors of ∆XYZ meet at M. Find MK. X L Z M 8 5 J K Y D C