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In the given figure, you are asked to find the measures of angles \( m \angle 1 \) and \( m \angle 2 \), given that \( a \parallel b \), which means lines \( a \) and \( b \) are parallel, and \( t \) is a transversal.

### Step-by-step Solution:

1. **Identifying corresponding angles**: Since \( a \parallel b \) and \( t \) is a transversal, angle 1 and the given \( 75^\circ \) angle are *corresponding angles*. Corresponding angles are equal when a transversal crosses parallel lines. Thus:
   \[
   m \angle 1 = 75^\circ
   \]

2. **Identifying alternate interior angles**: Angle 2 and the given \( 75^\circ \) angle are *alternate interior angles*. Alternate interior angles are also congruent when two lines are parallel. Therefore:
   \[
   m \angle 2 = 75^\circ
   \]

### Final Answer:
- \( m \angle 1 = 75^\circ \)
- \( m \angle 2 = 75^\circ \)

### Justification:
- **Corresponding angles** are congruent when the transversal intersects parallel lines.
- **Alternate interior angles** are congruent when the transversal intersects parallel lines.

Would you like further clarification on any of the steps or a deeper explanation?

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Here are 5 questions that expand on the topic:

1. What are other pairs of corresponding angles in this diagram?
2. How would the angle measures change if the parallel lines were not given?
3. What is the relationship between supplementary angles in this diagram?
4. How do alternate exterior angles behave with parallel lines and transversals?
5. How can you prove that two lines are parallel using angle relationships?

**Tip**: In a transversal, the sum of consecutive interior angles is always \(180^\circ\).

In each figure below, find m∠1 and m∠2, if a ∥ b. Justify your solution with reasons.

In this problem, we find the measures of angles formed by a transversal intersecting parallel lines. Using the Corresponding Angles Postulate and the Alternate Interior Angles Theorem, the solution shows that m∠1 and m∠2 both equal 75°. Suitable for students in Grades 7-9.

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