Write a two column proof.

Given: angle STV is congruent to angle TVU, angle STU is congruent to angle UVS.

Prove: Angle SVT is congruent to angle UTV.

To write a two-column proof, we need to present the statements and the corresponding reasons for each step. Here's how you can write a two-column proof for proving that Angle SVT is congruent to angle UTV:

| Statements | Reasons |
|-------------------|---------------------|
| 1. angle STV is congruent to angle TVU | Given |
| 2. angle STU is congruent to angle UVS | Given |
| 3. angle STU + angle UVS = angle SVT + angle UTV | Angle addition postulate |
| 4. angle STU + angle UVS = angle STV + angle TVU | Substitution (step 1) |
| 5. angle STV + angle TVU = angle SVT + angle UTV | Addition property of equality (step 4) |
| 6. angle SVT + angle UTV = angle STU + angle UVS | Commutative property (step 5) |
| 7. angle STU + angle UVS = angle SVT + angle UTV | Transitive property (step 6) |
| 8. angle SVT is congruent to angle UTV | Subtraction property (step 3) |

Using the given information and basic properties of equality and addition, we can deduce that angle SVT is congruent to angle UTV.

To prove: Angle SVT is congruent to angle UTV.

Statement | Reason
-----------------------------------------------------------------
1. angle STV is congruent to angle TVU | Given
2. angle STU is congruent to angle UVS | Given
3. angle STU is congruent to angle STV | Congruent angles are transitive
4. angle UVS is congruent to angle TVU | Congruent angles are transitive
5. angle SVT is congruent to angle UTV | Angle addition postulate (angle STU + angle STV = angle SVT; angle UVS + angle TVU = angle UTV)
6. Angle SVT is congruent to angle UTV | Substitution (from statements 3, 4, and 5)

Draw a diagram.

SVU = SVT + TVU
STU = STV + UTV

So,

SVT = SVU - TVU
UTV = STU - STV

Since SVU = STU and TVU = STV,

SVT = STU - STV = UTV