https://file.aiquickdraw.com/mathbot/file/1726410552798qk3qa139.jpg

The image you uploaded presents a geometric diagram with a proof related to the perpendicularity of lines \( SU \) and \( RT \). Let’s analyze the proof:

### Given Information:
- The goal is to prove that \( SU \perp RT \).
- The diagram shows two intersecting lines \( SU \) and \( RT \) at point \( V \).
- Angles \( \angle RVS \) and \( \angle SVT \) are involved in the proof.

### Proof Structure:
1. **Step 1:** \( \angle RVS \cong \angle SVT \)
   - This suggests that angles \( \angle RVS \) and \( \angle SVT \) are congruent, possibly by the vertical angle theorem (vertical angles are congruent).

2. **Step 2:** \( m\angle RVS + m\angle SVT = 180^\circ \)
   - This is the **linear pair postulate**: if two angles form a straight line, their measures add up to \( 180^\circ \).

3. **Step 3:** \( m\angle RVS + m\angle RVS = 180^\circ \)
   - Since \( \angle RVS \cong \angle SVT \), both angles have the same measure. Hence, this equation simplifies to \( m\angle RVS + m\angle RVS = 180^\circ \).

4. **Step 4:** \( m\angle RVS = 90^\circ \)
   - Dividing both sides of the equation in step 3 by 2, we find that \( m\angle RVS = 90^\circ \).

5. **Step 5:** \( SU \perp RT \)
   - Since \( m\angle RVS = 90^\circ \), the lines \( SU \) and \( RT \) are perpendicular by definition, because a right angle is formed at their intersection.

### Completed Proof:
- **Conclusion:** \( SU \perp RT \), as the angle between them is \( 90^\circ \).

Let me know if you would like any additional clarifications!

### Relative Questions:
1. What is the vertical angle theorem, and how does it apply here?
2. Can you explain the linear pair postulate more in-depth?
3. Why does proving \( \angle RVS = 90^\circ \) guarantee that \( SU \perp RT \)?
4. How does symmetry play a role in the congruency of angles in this proof?
5. What other methods could be used to prove lines are perpendicular?

### Tip:
Always look for the properties of special angles (like vertical angles or linear pairs) when dealing with intersecting lines in geometric proofs.

Explore the proof that SU is perpendicular to RT using the vertical angle theorem and linear pair postulate. Learn how congruent angles and right-angle formation confirm perpendicularity in geometric figures. Suitable for Grades 9-10.

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