GEOMETRY MODULE 2 LESSON 18 SIMILARITY AND THE ANGLE BISECTOR THEOREM OPENING EXERCISE What is an angle bisector? The bisector of an angle is a ray in the interior of the angle such that the two adjacent angles formed by it have equal measures. Describe the angle relationships formed when parallel lines are cut by a transversal. When parallel lines are cut by a transversal, the corresponding, alternate interior, and alternate exterior angles are all congruent; the same-side interior angles are supplementary. What are the properties of an isosceles triangle? An isosceles triangle has at least two congruent sides, and the angles opposite to the congruent sides (base angles) are also congruent. In the diagram below, the angle bisector of ∠퐴 in ∆퐴퐵퐶 meets side 퐵퐶̅̅̅̅ at point D. Does the angle bisector create any observable relationships with respect to the side lengths of the triangle? 퐵퐷 퐵퐴 = = 2 = 푠푐푎푙푒 푓푎푐푡표푟 퐶퐷 퐶퐴 MOD2 L18 1 The Angle Bisector Theorem: In ∆퐴퐵퐶, if the angle 퐵퐷 퐵퐴 bisector of ∠퐴 meets side 퐵퐶̅̅̅̅ at point D, then = . 퐶퐷 퐶퐴 (The bisector of an angle of a triangle splits the opposite side into segments that have the same ratio as the adjacent sides.) Proof of the Angle Bisector Theorem Consider the following construction where the line through vertex C is parallel to side 퐴퐵̅̅̅̅. We will prove the Angle Bisector Theorem via Similar Triangles. ∠퐷퐸퐶 ≅ ∠퐵퐴퐷 Alternate interior angles are congruent. ∠퐴퐷퐵 ≅ ∠퐸퐷퐶 Vertical angles are congruent. ∆퐴퐵퐷 ~ ∆퐸퐶퐷 AA Criterion 퐵퐷 퐵퐴 = Corresponding sides of similar triangles are proportional. 퐶퐷 퐶퐸 퐵퐷 퐵퐴 However, we must show = . Consider the ∆퐴퐶퐸. What can be concluded about the 퐶퐷 퐶퐴 triangle that can help us in our proof. Since angle A was bisected and ∠퐷퐸퐶 ≅ ∠퐵퐴퐷, we can conclude ∠퐷퐸퐶 ≅ ∠퐶퐴퐷. Therefore, ∆퐴퐶퐸 is an isosceles triangle. From this we conclude, 퐶퐴 = 퐶퐸. Through substitution, 퐵퐷 퐵퐴 = . The theorem is proved. 퐶퐷 퐶퐴 MOD2 L18 2 PRACTICE 1. The sides of a triangle are 8, 12, and 15. An angle bisector meets the side of length 15. Find the lengths x and y. Justify your work with a calculation and/or statement. 2. The sides of a triangle are 8, 12, and 15. An angle bisector meets the side of length 12. Find the lengths x and y. Justify your work with a calculation and/or statement. MOD2 L18 3 3. The angle bisector of an angle splits the opposite side of a triangle into lengths 5 and 6. The perimeter of the triangle is 33. Find the lengths of the other two sides. ON YOUR OWN 1. The sides of a triangle are 12, 16, and 21. An angle bisector meets the side of length 21. Find the lengths x and y. Justify your work with a calculation and/or statement. 2. The perimeter of ∆푈푉푊 is 22 ½ . 푊푍⃗⃗⃗⃗⃗⃗ bisects ∠푈푊푉, 푈푍 = 2, 1 and 푉푍 = 2 . Find UW and VW. 2 MOD2 L18 4 .