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Complete the two-column proof by filling in the blanks.

Prove that the triangles shown in the figure above are congruent.

**Statements**

**Reason**

1.

1. given

2.

2.         [ Select ]      ["Transitive Property", "Third angles theorem", "Definition of congruent triangles", "Reflexive property"]

3.         [ Select ]      ["Z", "ZWV", "ZVW"]  ,         [ Select ]      ["Z", "ZVW", "ZWV"]

3. given

4.

4.         [ Select ]      ["Transitive Property", "Definition of congruent triangles", "Reflexive property", "Third angles theorem"]

5.

5.         [ Select ]      ["Reflexive property", "Third angles theorem", "Transitive Property", "Definition of congruent triangles"]

To complete the two-column proof for proving that triangles \( \triangle XVW \) and \( \triangle ZVW \) in the figure are congruent, let’s analyze the diagram and go through each step with the appropriate statements and reasons.

### Analysis
From the diagram, we observe that:
1. \( XV = ZV \) and \( XW = ZW \) (indicated by tick marks).
2. \( \angle XVW \) is congruent to \( \angle ZVW \) (indicated by the arc marks).
3. The side \( VW \) is shared between both triangles.

Given these observations, we can now complete the two-column proof.

---

### Proof
**Statements**

| **Statement**                             | **Reason**                                     |
|------------------------------------------|------------------------------------------------|
| 1. \( XV = ZV \), \( XW = ZW \), \( \angle XVW \cong \angle ZVW \) | 1. Given |
| 2. \( VW = VW \)                          | 2. Reflexive property                           |
| 3. \( \triangle XVW \) and \( \triangle ZVW \) | 3. Given                                      |
| 4. Triangles \( \triangle XVW \cong \triangle ZVW \) | 4. Definition of congruent triangles (using the Side-Angle-Side postulate) |
| 5. \( \triangle XVW \cong \triangle ZVW \) | 5. Definition of congruent triangles            |

Each selection fits as follows:

1. Statement: \( XV = ZV \), \( XW = ZW \), \( \angle XVW \cong \angle ZVW \); Reason: **Given**
2. Statement: \( VW = VW \); Reason: **Reflexive property**
3. Statement: Triangles \( \triangle XVW \) and \( \triangle ZVW \); Reason: **Given**
4. Reason: **Definition of congruent triangles**
5. Reason: **Definition of congruent triangles**

Would you like further details on this proof? 

### Questions to Deepen Understanding
1. How can the Side-Angle-Side (SAS) postulate be used to prove congruence in other triangle pairs?
2. Why is the reflexive property essential in this proof?
3. Can other triangle congruence postulates, like ASA or AAS, be used here? Why or why not?
4. What does the term "definition of congruent triangles" specifically imply in a proof?
5. How would we approach this proof if we were asked to use transformations?

### Tip
Remember that the reflexive property (an object being equal to itself) is very useful in geometry, especially for proving congruence in triangles that share a side.

Complete the two-column proof by filling in the blanks.

Prove that the triangles shown in the figure above are congruent.

**Statements**

**Reason**

1.

1. given

2.

2.     [ Select ]     ["Transitive Property", "Third angles theorem", "Definition of congruent triangles", "Reflexive property"]

3.     [ Select ]     ["Z", "ZWV", "ZVW"] ,     [ Select ]     ["Z", "ZVW", "ZWV"]

3. given

4.

4.     [ Select ]     ["Transitive Property", "Definition of congruent triangles", "Reflexive property", "Third angles theorem"]

5.

5.     [ Select ]     ["Reflexive property", "Third angles theorem", "Transitive Property", "Definition of congruent triangles"]

Learn how to complete a two-column proof to show that triangles in a figure are congruent. This problem uses the Side-Angle-Side (SAS) Postulate and the Reflexive Property, and is suitable for students in Grades 9-10.

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