Let's begin by analyzing the given information and the desired outcome of our proof. We want to show that △ WXZ is congruent to △ YXZ. Recall that by the definition of congruent polygons we want to show that the sides and the angles of these triangles are congruent.
Congruent sides
Congruent angles
WX ≅ YX
∠ WXZ ≅ ∠ YXZ
WZ ≅ YZ
∠ XZW ≅ ∠ XZY
XZ ≅ XZ
∠ ZWX ≅ ∠ ZYX
We are given that WX ≅ YX, WZ ≅ YZ, ∠ WXZ ≅ ∠ YXZ, and ∠ XZW ≅ ∠ XZY. Therefore, we need to prove that XZ ≅ XZ and ∠ ZWX ≅ ∠ ZYX.
Notice that the triangles share the side XZ, and by the Reflexive Property of Congruence we know that XZ ≅ XZ.
Statement
By the Reflexive Property of Congruence, XZ ≅ XZ.
Next, we know that ∠ WXZ ≅ ∠ YXZ and ∠ XZW ≅ ∠ XZY. Two angles of △ WXZ are congruent to two angles of △ YXZ. Therefore, by the Third Angles Theorem we can conclude that the third angles are congruent, ∠ ZWX ≅ ∠ ZYX.
Statement
By the Third Angles Theorem, ∠ ZWX ≅ ∠ ZYX.
We have shown that all the sides and the angles in the triangles are congruent.
Congruent sides
Congruent angles
WX ≅ YX
∠ WXZ ≅ ∠ YXZ
WZ ≅ YZ
∠ XZW ≅ ∠ XZY
XZ ≅ XZ
∠ ZWX ≅ ∠ ZYX
Therefore, by the definition of congruent polygons, △ WXZ ≅ △ YXZ.
Statement
By the definition of congruent polygons, △ WXZ ≅ △ YXZ.
Completed Proof
Given:& ∠ WXZ ≅ ∠ YXZ, ∠ XZW ≅ ∠ XZY, & WX ≅ YX, WZ ≅ YZ
Prove:& △ WXZ ≅ △ YXZ
Proof: By the Reflexive Property of Congruence, XZ ≅ XZ. By the Third Angles Theorem, ∠ ZWX ≅ ∠ ZYX. By the definition of congruent polygons, △ WXZ ≅ △ YXZ.
