Let's begin by analyzing the given information and the desired outcome of our proof. We want to show that â–³ WXZ is congruent to â–³ 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.
