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Objective: Identify the triangle congruency theorem that can be used to prove two triangles congruent.

Given: ∠XWU ≅ ∠ZVT; ∠ZTV ≅ ∠XUW; TU ≅ VW

Which relationship in the diagram is true?

A.     △XYZ ≅ △XYS by SSS

B.     △ZYX ≅ △VYU by AAS

C.     △RYZ ≅ △XZY by SAS

D.     △ZVT ≅ △XWU by ASA

The image contains a geometric figure with several triangles labeled, and the problem asks to identify the correct triangle congruency theorem based on the given relationships. The given information is:

- ∠XWU ≅ ∠ZVT (Angles are congruent)
- ∠ZTV ≅ ∠XUW (Angles are congruent)
- TU ≅ VW (Sides are congruent)

Let's analyze each answer choice and the congruency theorems.

### ASA (Angle-Side-Angle):
This theorem states that two triangles are congruent if two angles and the included side of one triangle are congruent to the corresponding parts of another triangle.

### SAS (Side-Angle-Side):
This theorem states that two triangles are congruent if two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle.

### SSS (Side-Side-Side):
This theorem states that two triangles are congruent if all three sides of one triangle are congruent to the corresponding sides of another triangle.

### AAS (Angle-Angle-Side):
This theorem states that two triangles are congruent if two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle.

### Solution:
The given information (two pairs of congruent angles and one congruent side) fits the **ASA (Angle-Side-Angle)** theorem, as two angles and the included side are congruent between triangles ZVT and XWU.

Thus, the correct answer is:

**D. △ZVT ≅ △XWU by ASA**

Would you like further clarification or details?

Here are five related questions to expand on this topic:
1. What is the difference between ASA and AAS congruence theorems?
2. How does the SAS congruence theorem apply in different geometric figures?
3. Can two triangles be congruent if only two sides and no angles are congruent?
4. Why is the included side important in the ASA and SAS theorems?
5. How can you use congruent triangles to find unknown angles or side lengths?

**Tip:** When analyzing triangle congruency, always check if the congruent parts (angles and sides) match the conditions for the appropriate congruence theorem, like ASA or SAS.

Objective: Identify the triangle congruency theorem that can be used to prove two triangles congruent.

Given: ∠XWU ≅ ∠ZVT; ∠ZTV ≅ ∠XUW; TU ≅ VW

Which relationship in the diagram is true?

A. △XYZ ≅ △XYS by SSS

B. △ZYX ≅ △VYU by AAS

C. △RYZ ≅ △XZY by SAS

D. △ZVT ≅ △XWU by ASA

Explore how to prove that triangles ZVT and XWU are congruent using the ASA (Angle-Side-Angle) Congruency Theorem. This detailed explanation includes an analysis of the congruency conditions applied to geometric figures, making it ideal for high school students learning advanced geometry.

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