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To complete the proof that \(\angle RST \cong \angle UVW\), we need to reason through the congruence of the two angles, based on the given information.

### Step-by-Step Completion:

We are provided with the following information:
- \(ST \perp RS\) and \(VW \perp UV\).
- \(m\angle RST = 90^\circ\) and \(m\angle UVW = 90^\circ\) by the definition of perpendicular lines.

Now let's complete the proof.

### Given Proof:
1. \(ST \perp RS\) &emsp;---&emsp;Given  
2. \(VW \perp UV\) &emsp;---&emsp;Given  
3. \(m\angle RST = 90^\circ\) &emsp;---&emsp;Definition of perpendicular lines  
4. \(m\angle UVW = 90^\circ\) &emsp;---&emsp;Definition of perpendicular lines  

Now, to finish the proof, we need to use **Angle Congruence** to conclude that the two right angles are congruent.

5. \(\angle RST \cong \angle UVW\) &emsp;---&emsp;**All right angles are congruent**  

Thus, \(\angle RST \cong \angle UVW\) because all right angles are congruent by definition.

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Would you like any further details or explanation on any specific step?

### 5 Related Questions:
1. How do perpendicular lines guarantee 90-degree angles?
2. Why are all right angles congruent?
3. What is the definition of angle congruence in geometry?
4. Can you explain how congruent angles are used in geometric proofs?
5. What is the significance of using "given" statements in proofs?

### Tip:
When proving angle congruence in geometric proofs, always remember that right angles are automatically congruent, a property you can apply frequently!

Explore how to prove that two angles are congruent when two pairs of lines are perpendicular, with a focus on the definition of perpendicular lines and the congruence of right angles. This topic is suitable for middle school students studying geometry.

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