Use the definition of an angle bisector. Also, notice that â–³ XYW and â–³ ZYW have a side in common.
See solution.
Practice makes perfect
Since △ XYZ is equilateral, we get XY ≅ ZY ≅ XZ.
We are told that WY bisects ∠XYZ, which means that ∠XYW ≅ ∠ZYW. Additionally, WY is a common side for both △ XYW and △ ZYW.
By the Reflexive Property of Congruent Segments we have WY ≅ WY.
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XY ≅ ZY & Side
∠XYW ≅ ∠ZYW & Included Angle
YW ≅ YW & Side
By applying the Side-Angle-Side (SSS) Congruence Postulate we get △ XYW ≅ △ ZYW, and so XW≅ ZW. We can summarize this proof in a paragraph as follows.
Completed Proof
Given: & â–³ XYZ is equilateral
& WY bisects ∠XYZ
Prove: & XW≅ ZW
Proof: Since △ XYZ is equilateral we get XY ≅ ZY. Next, WY is a common side for both △ XYW and △ ZYW, and by the Reflexive Property of Congruent Segments we get WY ≅ WY. Besides, WY bisects ∠XYZ, which implies ∠XYW ≅ ∠ZYW. Thus, by the Side-Angle-Side (SSS) Congruence Postulate we get △ XYW ≅ △ ZYW and so, XW≅ ZW.
