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This image presents a geometric proof to demonstrate that line **SU** is perpendicular to line **TV**. The table lists statements and their corresponding reasons, with some reasons missing. Here's how to complete the proof based on the given diagram and steps:

### Proof:
We are given two angles, \( \angle TWU \) and \( \angle SWT \), and we need to show that \( SU \perp TV \).

#### Step-by-Step Completion:

1. **Statement 1**: \( \angle TWU \cong \angle SWT \)  
   **Reason**: Given (This means the two angles are congruent).

2. **Statement 2**: \( m\angle SWT + m\angle TWU = 180^\circ \)  
   **Reason**: *Linear pair postulate* (Adjacent angles formed by intersecting lines that lie on the same line sum up to 180°).

3. **Statement 3**: \( m\angle SWT + m\angle SWT = 180^\circ \)  
   **Reason**: *Substitution property* (Since \( \angle TWU \cong \angle SWT \), we can substitute \( m\angle TWU \) with \( m\angle SWT \)).

4. **Statement 4**: \( m\angle SWT = 90^\circ \)  
   **Reason**: *Properties of addition, subtraction, multiplication, and division* (Dividing both sides of \( 2m\angle SWT = 180^\circ \) by 2 gives \( m\angle SWT = 90^\circ \)).

5. **Statement 5**: \( SU \perp TV \)  
   **Reason**: *Definition of perpendicular lines* (If an angle between two intersecting lines is 90°, then the lines are perpendicular).

This completes the proof that **SU** is perpendicular to **TV**.

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Let me know if you want more details or have any questions!

Here are 5 related questions to deepen your understanding:
1. What is the difference between congruent and supplementary angles?
2. How can the linear pair postulate be applied to other geometric proofs?
3. What is the substitution property in geometry, and how is it used in proofs?
4. How do perpendicular lines relate to right angles?
5. Can two lines be perpendicular if none of the angles formed are 90 degrees?

**Tip**: Always ensure that each step of a proof logically follows from the previous step, and use geometric properties correctly!

This page demonstrates a step-by-step geometric proof that line SU is perpendicular to line TV. The problem involves concepts like linear pairs and the substitution property, suitable for students in Grades 8-10. Learn the key steps to prove perpendicularity in intersecting lines.

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