https://file.aiquickdraw.com/mathbot/file/1728922557196z0vztsec.jpg

Let's analyze and complete the proof provided in the image step by step.

We are given that:

- \( m \angle WOZ = 90^\circ \) (Given)
- We need to prove that \( m \angle WOX = 90^\circ \).

### Steps to complete the proof:

1. **Assumption**: Suppose \( m \angle WOX \neq 90^\circ \). This is the hypothesis we use for contradiction. This assumption leads us to consider that \( \angle WOX \) is supplementary to \( \angle WOZ \) (because these are adjacent angles on a straight line).
   
2. **Reasoning for supplementarity**: \( \angle WOX \) is supplementary to \( \angle WOZ \), meaning that the sum of their measures must be \( 180^\circ \).
   
   \[
   m \angle WOX + m \angle WOZ = 180^\circ
   \]
   
3. **Substitute the value of \( m \angle WOZ \)**: From the given information, \( m \angle WOZ = 90^\circ \), so:
   
   \[
   m \angle WOX + 90^\circ = 180^\circ
   \]
   
4. **Subtraction to find \( m \angle WOX \)**: By the subtraction property of equality:
   
   \[
   m \angle WOX = 180^\circ - 90^\circ = 90^\circ
   \]
   
5. **Contradiction**: This contradicts the assumption that \( m \angle WOX \neq 90^\circ \).

6. **Conclusion**: Therefore, the assumption is false, and it follows that \( m \angle WOX = 90^\circ \).

### Filling in the blanks from the image:

1. The first blank should be filled with **linear pair**, as \( \angle WOX \) and \( \angle WOZ \) form a linear pair.
   
2. The second blank should be filled with **equals**, since \( m \angle WOZ = 90^\circ \).

Would you like further details on this proof or clarification on any steps?

### Five relative questions:

1. What is the significance of using a contradiction in this proof?
2. How is the concept of a linear pair applied in geometric proofs?
3. Can you explain why the sum of supplementary angles is always \( 180^\circ \)?
4. What is the substitution property of equality in geometry?
5. How can we apply the subtraction property of equality to other angle-related problems?

### Tip:

Always start geometric proofs by identifying given information and the goal (what you're proving). This helps in outlining a clear strategy for the proof.

Given: m∠WOZ = 90°. Prove: m∠WOX = 90° using geometric principles.

Explore the geometric proof of m∠WOX = 90° by applying the Linear Pair Theorem and the concept of supplementary angles. Follow a step-by-step guide to complete the proof using properties of equality. Suitable for students in Grades 9-10.

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