Use the definition of angle bisector and look at the corresponding congruent parts.
See solution.
Practice makes perfect
We want to prove the following conjecture by writing a paragraph proof. Here is also what we know.
Given: & ∠ W ≅ ∠ Y, WZ ≅ YZ,
& XZ bisects ∠ WZY.
Prove: & △ XWZ ≅ △ XYZ
Let's focus on the diagram. By the definition of an angle bisector, we have that ∠ XZW and ∠ XZY are congruent. Also, we are told that ∠ W ≅ ∠ Y and WZ ≅ YZ. Let's mark this, as well as other corresponding congruent parts of both triangles, in the diagram.
There are two pairs of congruent angles and a pair of congruent sides in △ XWZ and △ XYZ.
cc
∠ XZW ≅ ∠ XZY & Angle
WZ ≅ YZ & Included Side
∠ W ≅ ∠ Y & Angle
Two angles and a non-included side of △ XWZ are congruent to two angles and a non-included side of △ XYZ. By the Angle-Angle-Side (AAS) Congruence Postulate we can conclude that the triangles are congruent.
△ XWZ ≅ △ XYZ
Paragraph Proof
We can now summarize our findings in a paragraph proof.
