# Similarity and the Angle Bisector Theorem

### New York State Common Core Math Geometry, Module 2, Lesson 18

Student Outcomes

• Students state, understand, and prove the Angle Bisector Theorem.
• Students use the Angle Bisector Theorem to solve problems.

Similarity and the Angle Bisector Theorem

Classwork

Opening Exercise

a. What is an angle bisector?
b. Describe the angle relationships formed when parallel lines are cut by a transversal.
c. What are the properties of an isosceles triangle?

Discussion

In the diagram below, the angle bisector of β π΄ in β³ π΄π΅πΆ meets side π΅πΆΜΜΜΜ at point π·. Does the angle bisector create any observable relationships with respect to the side lengths of the triangle

Exercises 1β4

1. The sides of a triangle are 8, 12, and 15. An angle bisector meets the side of length 15. Find the lengths π₯ and π¦. Explain how you arrived at your answers.
2. The sides of a triangle are 8, 12, and 15. An angle bisector meets the side of length 12. Find the lengths π₯ and π¦.
3. The sides of a triangle are 8, 12, and 15. An angle bisector meets the side of length 8. Find the lengths π₯ and π¦.
4. 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.

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