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To prove that \( m \angle 1 = m \angle 2 \), we will proceed step by step using the given information and properties of angles.

### Step-by-step Proof:

1. **Given:**
   - \( m \angle QPS = m \angle TPV \)
   - \( m \angle 3 = m \angle 4 \)
   
2. **Interpretation of the Diagram:**
   - From the diagram, we can see that angles \( \angle 1 \), \( \angle 3 \), and \( \angle QPS \) form a straight line around point \( P \), and similarly, angles \( \angle 2 \), \( \angle 4 \), and \( \angle TPV \) also form a straight line around point \( P \).

3. **Step 1: Apply the given equalities**:
   - The statement \( m \angle QPS = m \angle TPV \) tells us that these two large angles are congruent.
   - The equality \( m \angle 3 = m \angle 4 \) tells us that these smaller interior angles within the larger angles are also congruent.

4. **Step 2: Use the angle subtraction property**:
   - Since \( m \angle QPS = m \angle 1 + m \angle 3 \) and \( m \angle TPV = m \angle 2 + m \angle 4 \), we can substitute from the second given statement.
   - This gives us \( m \angle QPS = m \angle 1 + m \angle 3 \) and \( m \angle TPV = m \angle 2 + m \angle 3 \) (since \( m \angle 3 = m \angle 4 \)).

5. **Step 3: Solve for \( m \angle 1 \) and \( m \angle 2 \)**:
   - Using the fact that \( m \angle QPS = m \angle TPV \), we now have:
     \[
     m \angle 1 + m \angle 3 = m \angle 2 + m \angle 3
     \]
   - Subtract \( m \angle 3 \) from both sides:
     \[
     m \angle 1 = m \angle 2
     \]

Thus, we have proved that \( m \angle 1 = m \angle 2 \).

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Would you like further details on any part of this proof? Here are five related questions to explore further:

1. Why are angles formed around a point always equal to 360 degrees?
2. How does the angle addition postulate apply in this situation?
3. What is the importance of congruent angles in geometric proofs?
4. How could this proof be extended to other types of angle relationships?
5. What other methods can be used to prove angles are congruent in geometry?

**Tip**: When proving equal angles, always check if you can apply properties of adjacent or supplementary angles.

Use the given information to prove that m∠1 = m∠2. Given: m∠QPS = m∠TPV, m∠3 = m∠4. Prove: m∠1 = m∠2.

Learn how to prove that m∠1 = m∠2 using given information about congruent angles and the angle subtraction property. This problem involves geometry concepts such as angle addition and congruent angles. Suitable for students in Grades 7-10.

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