Question

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

Explanation:

To determine the correct triangle congruence, we analyze each option:

1. Option 1: $\boldsymbol{\triangle XYZ \cong \triangle XYS}$ by SSS

We need three pairs of congruent sides. The diagram doesn’t show enough information to confirm all three sides of $\triangle XYZ$ and $\triangle XYS$ are congruent. Eliminate this.

1. Option 2: $\boldsymbol{\triangle ZYX \cong \triangle VYU}$ by AAS

AAS requires two angles and a non - included side. The given information and the diagram don’t support that these two triangles have the necessary congruent angles and non - included side. Eliminate this.

1. Option 3: $\boldsymbol{\triangle RYZ \cong \triangle XZY}$ by SAS

SAS requires two sides and the included angle. There’s no clear evidence from the diagram or given info that these two triangles have the required congruent sides and included angle. Eliminate this.

1. Option 4: $\boldsymbol{\triangle ZVT \cong \triangle XWU}$ by ASA

ASA requires two angles and the included side. We know:

• $\angle XWU \cong \angle ZVT$ (given),
• $\overline{TU} \cong \overline{VW}$, and since $TU + UV=VW + UV$ (segment addition), $\overline{TV} \cong \overline{WU}$ (included side),
• $\angle ZTV \cong \angle XUW$ (given).

So, by ASA, $\triangle ZVT \cong \triangle XWU$.

Answer:

$\boldsymbol{\triangle ZVT \cong \triangle XWU}$ by ASA