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The proof you're working on involves showing that the sum of three angles \( m \angle Q + m \angle U + m \angle QRU = 180^\circ \) without using the Triangle Angle Sum Theorem.

Let's examine the steps in the proof, filling in the missing reasons:

### Steps in the Proof

| **Statement**                             | **Reason**                                   |
|-------------------------------------------|---------------------------------------------|
| 1. \( QU \parallel RT \)                    | Given                                        |
| 2. \( \angle Q \cong \angle SRT \)         | **Corresponding Angles Postulate**           |
| 3. \( \angle U \cong \angle TRU \)         | **Corresponding Angles Postulate**           |
| 4. \( m \angle SRU = m \angle SRT + m \angle TRU \) | Additive Property of Angle Measure            |
| 5. \( m \angle SRU = m \angle Q + m \angle U \) | Substitution (from Steps 2 and 3)             |
| 6. \( m \angle SRU + m \angle QRU = 180^\circ \)  | Angles forming a linear pair sum to \( 180^\circ \) |
| 7. \( m \angle Q + m \angle U + m \angle QRU = 180^\circ \) | Substitution (from Step 5)                      |

### Explanation:
1. **Corresponding Angles Postulate** is used in Steps 2 and 3 because \( QU \parallel RT \), and corresponding angles are congruent when a transversal cuts two parallel lines.
2. From Step 4, the Additive Property of Angle Measure is used because the whole angle \( \angle SRU \) is the sum of its parts.
3. In Step 5, the previously established congruent angles are substituted.
4. The key fact used in Step 6 is that \( \angle SRU \) and \( \angle QRU \) form a linear pair, which always sum to \( 180^\circ \).

Would you like more details on any specific step?

### Follow-Up Questions:
1. How is the Corresponding Angles Postulate applied in geometry?
2. What is a linear pair of angles, and why do they always sum to \( 180^\circ \)?
3. How can you use parallel lines to determine congruent angles in other problems?
4. Can the same reasoning apply to non-parallel lines? Why or why not?
5. What other properties of angles can be useful in triangle proofs?

### Tip:
When dealing with parallel lines and a transversal, always check for corresponding, alternate interior, and consecutive interior angles—these relationships often simplify angle calculations.

Complete the proof that m∠Q + m∠U + m∠QRU = 180° without using the Triangle Angle Sum Theorem.

Explore the step-by-step proof that the sum of angles m∠Q + m∠U + m∠QRU equals 180° without using the Triangle Angle Sum Theorem. This problem uses the Corresponding Angles Postulate, the Additive Property of Angle Measure, and properties of linear pairs. Suitable for high school students in Grades 9-12.

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