https://file.aiquickdraw.com/mathbot/file/17301429232118pyvsitg.jpg

To prove that \( \angle 1 \cong \angle 3 \), let's go through the steps logically, based on the given information:

### Given:
1. \( \angle 2 \cong \angle 4 \)
2. \( \angle 2 \) and \( \angle 3 \) are supplementary.

### Prove:
\( \angle 1 \cong \angle 3 \).

### Proof:
1. **Given**: \( \angle 2 \cong \angle 4 \).
   - Since \( \angle 2 \) is congruent to \( \angle 4 \), they have the same measure.

2. **Given**: \( \angle 2 \) and \( \angle 3 \) are supplementary.
   - This means that \( m\angle 2 + m\angle 3 = 180^\circ \).

3. **Substitution**:
   - Since \( m\angle 2 = m\angle 4 \) (from step 1), we can substitute \( m\angle 4 \) in place of \( m\angle 2 \) in the equation from step 2.
   - Therefore, \( m\angle 4 + m\angle 3 = 180^\circ \).
   
4. **Consecutive Interior Angles**:
   - \( \angle 1 \) and \( \angle 4 \) form a pair of alternate interior angles along the transversal \( TP \), given that \( TV \) and \( QM \) are parallel.
   - Since alternate interior angles are congruent when lines are parallel, \( \angle 1 \cong \angle 4 \).

5. **Conclusion**:
   - Since \( \angle 4 \cong \angle 2 \) (by the given information) and \( \angle 2 \) is supplementary to \( \angle 3 \), we can deduce by transitive properties that \( \angle 1 \cong \angle 3 \).

Therefore, we have proven \( \angle 1 \cong \angle 3 \).

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Would you like further clarification or additional steps?

### Related Questions
1. Why are alternate interior angles congruent when lines are parallel?
2. How does the transitive property apply in geometric proofs?
3. What does it mean for angles to be supplementary?
4. How does substitution work in geometry proofs?
5. Why is it important to prove congruency in geometric shapes?

### Tip
Always look for relationships such as parallel lines and transversals in geometric diagrams, as they often imply angle congruence or supplementary angles.

Provide reasons for the proof given in the diagram: \( \angle 2 \cong \angle 4 \) and \( \angle 2 \) and \( \angle 3 \) are supplementary. Prove: \( \angle 1 \cong \angle 3 \).

Explore a geometric proof involving angle congruence in a parallel line setup, given \( \angle 2 \cong \angle 4 \) and supplementary angles. Learn how to prove \( \angle 1 \cong \angle 3 \) using properties of parallel lines and transversals. Suitable for Grade 8-10 students.

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