https://file.aiquickdraw.com/mathbot/file/1729474658104aemzjxdh.jpg

To complete the proof in the table, here are the necessary reasons for the missing steps:

- **Step 3 (Reason)**: The reason for the third statement \(\angle T \cong \angle SQT\) is also the **Corresponding Angles Theorem**, since \(\overline{PT} \parallel \overline{QS}\) and \(\overline{TQ}\) is a transversal.
  
- **Step 4 (Reason)**: The equation \(m\angle RQT = m\angle RQS + m\angle SQT\) comes from **Angle Addition Postulate**. This postulate states that if two angles are adjacent, their measures add up to the measure of the larger angle they form.

- **Step 5 (Reason)**: The equality \(m\angle RQT = m\angle P + m\angle T\) follows from **Substitution** because earlier in the proof, it is established that \(m\angle RQS = m\angle P\) and \(m\angle SQT = m\angle T\).

Now the completed proof reads as follows:

| Statement | Reason |
|-----------|--------|
| 1. \(\overline{PT} \parallel \overline{QS}\) | Given |
| 2. \(\angle P \cong \angle RQS\) | Corresponding Angles Theorem |
| 3. \(\angle T \cong \angle SQT\) | Corresponding Angles Theorem |
| 4. \(m\angle RQT = m\angle RQS + m\angle SQT\) | Angle Addition Postulate |
| 5. \(m\angle RQT = m\angle P + m\angle T\) | Substitution |
| 6. \(m\angle RQT + m\angle PQT = 180^\circ\) | Angles forming a linear pair sum to \(180^\circ\) |
| 7. \(m\angle P + m\angle T + m\angle PQT = 180^\circ\) | Substitution |

Would you like any further clarification on this proof?

Here are five related questions to expand on this topic:
1. What is the Corresponding Angles Theorem and how does it apply in parallel lines?
2. Can you explain the Angle Addition Postulate in more detail with another example?
3. How do linear pairs and the concept of supplementary angles relate to this proof?
4. What is the significance of parallel lines and transversals in geometric proofs?
5. What are some other ways to prove that the sum of angles in a triangle is \(180^\circ\)?

**Tip:** Always keep track of corresponding angles when working with parallel lines and transversals, as they often play a key role in proofs.

Complete the proof that m∠P + m∠T + m∠PQT = 180° without using the Triangle Angle Sum Theorem. The figure shows PT || QS with transversal lines, and corresponding angles formed by parallel lines.

Learn how to prove that m∠P + m∠T + m∠PQT = 180° using parallel lines and angle relationships. This geometry problem involves the Corresponding Angles Theorem and the Angle Addition Postulate, key concepts for students in Grades 9-11.

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