Question

complete the proof that $\triangle txz \cong \triangle vyw$.

(image of two triangles with points z, t, y, u, x, w, v)

statement	reason
2 $\angle z \cong \angle w$	given
3 $\overline{wx} \cong \overline{tz}$	given
4 $xz = yz + xy$	additive property of length
5 $wy = wx + xy$	additive property of length
6 $xz = wx + xy$	substitution
7 $wy = xz$	transitive property of equality
8

Explanation:

Step1: Identify Congruence Criterion

We have two angles and a side. From steps 1 ($\angle TXZ \cong \angle VYW$), 2 ($\angle Z \cong \angle W$), and 7 ($WY = XZ$), we can use the AAS (Angle - Angle - Side) congruence criterion.

Step2: Formulate the Statement

The triangles $\triangle TXZ$ and $\triangle VYW$ have two angles and a non - included side congruent. So the statement for step 8 should be $\triangle TXZ \cong \triangle VYW$ and the reason is AAS (Angle - Angle - Side) Congruence Criterion.

Answer:

Statement: $\triangle TXZ \cong \triangle VYW$; Reason: AAS (Angle - Angle - Side) Congruence Criterion