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To prove that \( \triangle TEM \cong \triangle PER \) in a two-column format, let's analyze the given information:

### Given:
1. \( \overline{TM} \cong \overline{PR} \)
2. \( \overline{TM} \parallel \overline{RP} \)

### To Prove:
\( \triangle TEM \cong \triangle PER \)

### Proof:

| **Statement**                           | **Reason**                               |
|-----------------------------------------|------------------------------------------|
| 1. \( \overline{TM} \cong \overline{PR} \)                   | Given                                      |
| 2. \( \overline{TM} \parallel \overline{RP} \)               | Given                                      |
| 3. \( \angle TEM \cong \angle PER \)                         | Alternate Interior Angles (parallel lines) |
| 4. \( \overline{EM} \cong \overline{ER} \)                   | Reflexive Property                         |
| 5. \( \triangle TEM \cong \triangle PER \)                   | SAS (Side-Angle-Side) Congruence Postulate |

### Explanation:
- **Step 1 & 2**: These are the given conditions in the problem.
- **Step 3**: Since \( \overline{TM} \parallel \overline{RP} \), angles \( \angle TEM \) and \( \angle PER \) are congruent by the Alternate Interior Angles Theorem.
- **Step 4**: Segment \( \overline{EM} \) is congruent to itself by the Reflexive Property.
- **Step 5**: Using the SAS Postulate (two sides and the included angle are congruent), we conclude that \( \triangle TEM \cong \triangle PER \).

Would you like further explanation on any of these steps?

### Related Questions:
1. How does the Alternate Interior Angles Theorem apply to parallel lines?
2. What is the SAS Congruence Postulate?
3. Why is the Reflexive Property used in geometric proofs?
4. How do parallel lines affect angle relationships in a triangle?
5. What are other congruence postulates besides SAS?

### Tip:
In proofs involving parallel lines, always look for corresponding or alternate interior angles, as they are often key to establishing angle congruences.

Given: TM ≅ PR; TM ∥ RP. Prove: ΔTEM ≅ ΔPER.

Learn how to prove the congruence of ΔTEM and ΔPER using the Side-Angle-Side (SAS) Congruence Postulate and parallel lines. This proof involves applying the Alternate Interior Angles Theorem and Reflexive Property, suitable for high school geometry students.

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