Question

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

Explanation:

Step1: Recall the ASA (Angle - Side - Angle) congruence criterion

Two triangles are congruent by ASA if two angles and the included side of one triangle are congruent to two angles and the included side of the other triangle.

Step2: Identify the given congruent angles and side

We are given that $\angle XWU\cong\angle ZVT$, $\angle ZTV\cong\angle XUW$, and $TU\cong VW$. Since $TU + UV=TV$ and $VW+UV = UW$ (by segment addition postulate), we can get $TV = UW$ (by substitution, because $TU\cong VW$ and $UV$ is common).

In $\triangle ZVT$ and $\triangle XWU$:

• $\angle ZVT\cong\angle XWU$ (given)
• $TV = UW$ (proven above)
• $\angle ZTV\cong\angle XUW$ (given)

So, by the ASA congruence criterion, $\triangle ZVT\cong\triangle XWU$.

Let's check the other options:

• For $\triangle XYZ\cong\triangle XYS$ by $SSS$: There is no information given about the sides $YZ$, $YS$, $XZ$ and $XS$ to confirm all three sides are congruent.
• For $\triangle ZYX\cong\triangle VYU$ by $AAS$: There is no proper correspondence of angles and sides to apply the AAS (Angle - Angle - Side) criterion.
• For $\triangle RYZ\cong\triangle XZY$ by $SAS$: There is no information about the sides $RY$, $XZ$ (or other relevant sides) and the included angles to apply the SAS (Side - Angle - Side) criterion.

Answer:

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