Question

yz ≅ xz and wz ≅ vz. complete the proof that △vyz ≅ △wxz.

statement
1 yz ≅ xz
2 wz ≅ vz
3 ∠vzy ≅ ∠wzx
4 △vyz ≅ △wxz

reason
given
given

Explanation:

Step1: Vertical angles are congruent

$\angle VZY$ and $\angle WZX$ are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent. So, $\angle VZY\cong\angle WZX$.

Step2: Use the Side - Angle - Side (SAS) congruence criterion

We have $\overline{YZ}\cong\overline{XZ}$ (given), $\overline{WZ}\cong\overline{VZ}$ (given), and $\angle VZY\cong\angle WZX$ (from Step 1). The Side - Angle - Side (SAS) congruence criterion states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. So, $\triangle VYZ\cong\triangle WXZ$ by SAS.

Answer:

1. Reason: Vertical Angles Theorem; 4. Reason: Side - Angle - Side (SAS) Congruence Criterion.