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The image provided shows a geometric proof table to prove that ∠RST is congruent to ∠UVW, given that \( \overline{UV} \perp \overline{VW} \) and \( \overline{ST} \perp \overline{RS} \).

Here is a step-by-step completion of the proof:

### Given Information:
- \( \overline{UV} \perp \overline{VW} \)
- \( \overline{ST} \perp \overline{RS} \)

### Goal:
Prove that \( \angle RST \cong \angle UVW \).

### Proof Table:

| Statement                     | Reason                            |
|--------------------------------|-----------------------------------|
| 1. \( \overline{UV} \perp \overline{VW} \) and \( \overline{ST} \perp \overline{RS} \) | Given                              |
| 2. \( \angle UVW = 90^\circ \) and \( \angle RST = 90^\circ \)      | Definition of perpendicular lines  |
| 3. \( m\angle UVW = 90^\circ \)                                     | Measure of right angle             |
| 4. \( m\angle RST = 90^\circ \)                                     | Measure of right angle             |
| 5. \( m\angle UVW = m\angle RST \)                                  | Transitive Property (equal measures) |
| 6. \( \angle RST \cong \angle UVW \)                                | Definition of congruent angles     |

### Explanation:

1. **Step 1**: The given information states that both pairs of lines are perpendicular.
2. **Step 2**: By the definition of perpendicular lines, the angles formed by these lines are right angles (i.e., 90°).
3. **Step 3 and 4**: We state the specific measure of these angles as 90°.
4. **Step 5**: Since both angles are 90°, by the transitive property, their measures are equal.
5. **Step 6**: Finally, if two angles have equal measures, they are congruent by definition.

Would you like further clarification or details on any step?

Here are five follow-up questions related to this proof:
1. Why does perpendicularity guarantee that the angles are 90°?
2. What is the definition of congruent angles?
3. Could this proof work for any pair of perpendicular lines?
4. How does the transitive property help in proving angle congruence?
5. Why is it important to show that the measures of both angles are the same?

**Tip**: Always remember that congruent angles have equal measures, but the reverse is also true—if two angles have the same measure, they are automatically congruent.

Complete the proof that ∠RST ≅ ∠UVW, given UV ⊥ VW and ST ⊥ RS.

Learn how to prove that ∠RST is congruent to ∠UVW with a step-by-step geometric proof, involving perpendicular lines, right angles, and the transitive property. This proof is suitable for students in Grades 8-10, covering key concepts in geometry.

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