Question

ng triangle congruence theorems
given: \\( \angle x w u \cong \angle z v t ; \angle z t v \cong \angle x u w ; \overline{t u} \cong \overline{v w} \\)
which relationship in the diagram is true?
\\( \triangle x y z \cong \triangle x y s \\) by sss
\\( \triangle z v t \cong \triangle x w u \\) by asa
\\( \triangle r y z \cong \triangle x z y \\) by sas
\\( \triangle z y x \cong \triangle v y u \\) by aas

Explanation:

Step1: Analyze each option

• For $\triangle XYZ\cong\triangle XYS$ by SSS: There is no information given about the side - side - side equality of these two triangles.
• For $\triangle ZVT\cong\triangle XWU$ by ASA: We are given $\angle XWU\cong\angle ZVT$ (one angle), $\angle ZTV\cong\angle XUW$ (another angle), and $\overline{TU}\cong\overline{VW}$ which is not the included side for the ASA criterion for these two triangles.
• For $\triangle RYZ\cong\triangle XZY$ by SAS: There is no information about the side - angle - side equality of these two triangles.
• For $\triangle ZYX\cong\triangle VYU$ by AAS:
• $\angle ZYX$ and $\angle VYU$ are vertical angles, so $\angle ZYX\cong\angle VYU$.
• We know that in a triangle, if two angles and a non - included side are equal. Here, we can assume some angle - angle - side relationships based on the given congruent angles in the problem's context (the given $\angle XWU\cong\angle ZVT$ and $\angle ZTV\cong\angle XUW$ can be used to derive other angle congruences in the sub - triangles).

Answer:

$\triangle ZYX\cong\triangle VYU$ by AAS