Question

given: ∠xwu ≅ ∠zvt; ∠ztv ≅ ∠xuw; \\(\overline{tu} \cong \overline{vw}\\) which relationship in the diagram is true? \\(\bigcirc\\) \\(\triangle xyz \cong \triangle xys\\) by sss \\(\bigcirc\\) \\(\triangle zyx \cong \triangle vyu\\) by aas \\(\bigcirc\\) \\(\triangle ryz \cong \triangle xzy\\) by sas \\(\bigcirc\\) \\(\triangle zvt \cong \triangle xwu\\) by asa

Explanation:

Step1: Recall triangle congruence criteria

• SSS (Side - Side - Side): All three sides of one triangle are equal to the corresponding sides of another triangle.
• AAS (Angle - Angle - Side): Two angles and a non - included side of one triangle are equal to the corresponding two angles and non - included side of another triangle.
• SAS (Side - Angle - Side): Two sides and the included angle of one triangle are equal to the corresponding two sides and included angle of another triangle.
• ASA (Angle - Side - Angle): Two angles and the included side of one triangle are equal to the corresponding two angles and included side of another triangle.

Step2: Analyze each option

• Option 1: $\triangle XYZ\cong\triangle XYS$ by SSS

There is no information given about the sides $YZ$, $YS$ and $XZ$, $XS$ to use the SSS criterion.

• Option 2: $\triangle ZYX\cong\triangle VYU$ by AAS

There is no information given about the sides $ZY$, $VY$ and $YX$, $YU$ to use the AAS criterion.

• Option 3: $\triangle RYZ\cong\triangle XZY$ by SAS

There is no information given about the sides $RY$, $XZ$ and the included angles to use the SAS criterion.

• Option 4: $\triangle ZVT\cong\triangle XWU$ by ASA

Given $\angle XWU\cong\angle ZVT$, $\angle ZTV\cong\angle XUW$ and $TU\cong VW$. Since $TU + UV=TV$ and $VW+UV = UW$ (by segment addition postulate), so $TV = UW$.
In $\triangle ZVT$ and $\triangle XWU$:
$\angle ZVT=\angle XWU$ (given), $TV = UW$ (proven), $\angle ZTV=\angle XUW$ (given). So, by ASA (Angle - Side - Angle) criterion $\triangle ZVT\cong\triangle XWU$.

Answer:

$\triangle ZVT\cong\triangle XWU$ by $ASA$