Let's begin by analyzing the given information. We are given that one side of △ YXZ is congruent to one side of △ WZX. This is how we will begin our proof.
Statement1)& YX ≅ WZ Reason1)& Given
We are also given that YX ∥ ZW. Therefore, if we draw the lines passing through these sides we will get two parallel lines, a and b. If we also draw a line passing through the side XZ, then we will get a transversal t that intersects these parallel lines.
Notice that ∠ ZXY and ∠ XZW are alternate interior angles. Lines a and b are parallel, thus by the Alternate Interior Angles Theorem ∠ ZXY ≅ ∠ XZW.
Statement 2)& ∠ ZXY ≅ ∠ XZW Reason 2)& Alternate Interior Angles & Theorem
Next, from the diagram we can tell that the triangles share the side XZ. By the Reflexive Property of Congruence we know that XZ ≅ ZX.
Statement 3)& XZ ≅ ZX Reason 3)& Reflexive Property & of Congruence
Notice that now we know that two sides and the included angle of △ YXZ are congruent to two sides and the included angle of △ WZX. Thus, by the SAS Theorem △ YXZ ≅ △ WZX.
Statement4)& △ YXZ ≅ △ WZX Reason4)& SAS Theorem
