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 â–³ XWZ and â–³ XYZ.
cc
∠XZW ≅ ∠XZY & Angle
WZ ≅ YZ & Included Side
∠W ≅ ∠Y & Angle
Two angles and a non-included side of â–³ XWZ are congruent to two angles and a non-included side of â–³ 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.
Proof: By the definition of an angle bisector, we have ∠XZW ≅ ∠XZY. Besides that, we are told that ∠W ≅ ∠Y and WZ ≅ YZ. Therefore, by the Angle-Side-Angle (ASA) Congruence Postulate we conclude that △ XWZ ≅ △ XYZ.
