   Chapter 5.6, Problem 35E Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

Chapter
Section Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

Use Theorem 5.6.3 to complete the proof of this theorem: “If the bisector of an angle of a triangle also bisects the opposite side, then the triangle is an isosceles triangle.” Given: Δ X Y Z ; Y W → bisects ∠ X Y Z ; W X ¯ ≅ W Z ¯ Prove: Δ X Y Z is isosceles (HINT: Use a proportion to show that Y X = Y Z . )

To determine

To prove:

ΔXYZ is isosceles.

Explanation

Given:

We need to prove, “If the bisector of an angle of a triangle also bisects the opposite side, then the triangle is an isosceles triangle”.

In a ΔXYZ, YW bisects XYZ; WX¯WZ¯.

Theorem used:

Angle bisector theorem:

If a ray bisects one angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the two sides that form the bisected angle.

Proof:

Consider a ΔXYZ.

Given that YW bisects XYZ.

By angle bisector theorem, we get

WXWZ=YXYZ … (1)

Given that WXx

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