Question

complete the proof that \\( \triangle v w x \cong \triangle u y x \\).

Explanation:

Step1: Identify triangle parts

We have two triangles, $\triangle VWX$ and $\triangle UYX$. Let's list the known congruent parts:

• $\angle VXW \cong \angle UXY$ (Statement 2)
• $\overline{XY} \cong \overline{WX}$ (Statement 4)
• $\angle Y \cong \angle W$ (Statement 5)

Step2: Determine congruence criterion

Looking at the angles and side, we have two angles and a side. Specifically, we have $\angle Y \cong \angle W$ (right angles or congruent angles), $\overline{XY} \cong \overline{WX}$ (side), and $\angle VXW \cong \angle UXY$ (angle). This matches the ASA (Angle - Side - Angle) congruence criterion, or also, since we have two angles and a side, it can be AAS (Angle - Angle - Side) as well. But let's check the parts:
In $\triangle UYX$ and $\triangle VWX$:

• $\angle Y \cong \angle W$ (Statement 5)
• $\overline{XY} \cong \overline{WX}$ (Statement 4)
• $\angle UXY \cong \angle VXW$ (Statement 2)

So by the ASA (or AAS) congruence postulate, the triangles are congruent.

Answer:

The reason for $\triangle VWX \cong \triangle UYX$ is the ASA (Angle - Side - Angle) Congruence Postulate (or AAS, since two angles and a non - included side also work here. But based on the given statements, with $\angle Y\cong\angle W$, $\overline{XY}\cong\overline{WX}$, and $\angle UXY\cong\angle VXW$, ASA is applicable as the side is between the two angles). So the reason is "ASA (Angle - Side - Angle) Congruence Postulate" (or "AAS (Angle - Angle - Side) Congruence Theorem" depending on the order of angles and side, but ASA is more direct here as the side is included between the two angles).