Question

given: \\( \angle xwu \cong \angle zvt; \angle ztv \cong \angle 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 \(XZ, XS\) being equal. So, this option is incorrect.

• Option 2: \(\triangle ZYX\cong\triangle VYU\) by AAS

There is no information given about the sides \(ZY, VY\) and \(YX, YU\) related to the angles in a way that satisfies AAS. So, this option is incorrect.

• Option 3: \(\triangle RYZ\cong\triangle XZY\) by SAS

There is no information given about the sides \(RY, XZ\) and \(YZ\) being used with an included angle. So, this option is incorrect.

• Option 4: \(\triangle ZVT\cong\triangle XWU\) by ASA

We are given \(\angle XWU\cong\angle ZVT\) (one angle), \(\overline{TU}\cong\overline{VW}\) (so \(TV = WU\) since \(TV=TU + UV\) and \(WU = VW+UV\)), and \(\angle ZTV\cong\angle XUW\) (the second angle). The side \(TV\) (or \(WU\)) is the included side between the two given angles. So, by ASA (Angle - Side - Angle) congruence criterion, \(\triangle ZVT\cong\triangle XWU\)

Answer:

\(\triangle ZVT\cong\triangle XWU\) by \(ASA\) (the fourth option)