Download geometry module 2 lesson 18 similarity and the angle bisector

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.

̅̅̅̅ at point D. Does the
In the diagram below, the angle bisector of ∠𝐴 in ∆𝐴𝐵𝐶 meets side 𝐵𝐶
angle bisector create any observable relationships with respect to the side lengths of the
triangle?
𝐵𝐷 𝐵𝐴
=
= 2 = 𝑠𝑐𝑎𝑙𝑒 𝑓𝑎𝑐𝑡𝑜𝑟
𝐶𝐷 𝐶𝐴
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The Angle Bisector Theorem: In ∆𝐴𝐵𝐶, if the angle
𝐵𝐷
̅̅̅̅ at point D, then
bisector of ∠𝐴 meets side 𝐵𝐶
=
𝐶𝐷
𝐵𝐴
𝐶𝐴
.
(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 ̅̅̅̅
𝐴𝐵 . We will
prove the Angle Bisector Theorem via Similar Triangles.

∠𝐷𝐸𝐶 ≅ ∠𝐵𝐴𝐷 Alternate interior angles are congruent.

∠𝐴𝐷𝐵 ≅ ∠𝐸𝐷𝐶 Vertical angles are congruent.

∆𝐴𝐵𝐷 ~ ∆𝐸𝐶𝐷

𝐵𝐷
𝐶𝐷
=
𝐵𝐴
AA Criterion
Corresponding sides of similar triangles are proportional.
𝐶𝐸
𝐵𝐷
𝐵𝐴
However, we must show 𝐶𝐷 = 𝐶𝐴 . Consider the ∆𝐴𝐶𝐸. What can be concluded about the
triangle that can help us in our proof.
Since angle A was bisected and ∠𝐷𝐸𝐶 ≅ ∠𝐵𝐴𝐷, we can conclude ∠𝐷𝐸𝐶 ≅ ∠𝐶𝐴𝐷. Therefore,
∆𝐴𝐶𝐸 is an isosceles triangle. From this we conclude, 𝐶𝐴 = 𝐶𝐸. Through substitution,
𝐵𝐷
𝐶𝐷
=
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𝐵𝐴
𝐶𝐴
. The theorem is proved.
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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.
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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.
⃗⃗⃗⃗⃗⃗ bisects ∠𝑈𝑊𝑉, 𝑈𝑍 = 2,
2. The perimeter of ∆𝑈𝑉𝑊 is 22 ½ . 𝑊𝑍
1
and 𝑉𝑍 = 2 2. Find UW and VW.
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