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### Problem 7
Given \( s \parallel t \) and \( m\angle 7 = 84^\circ \), find \( m\angle 10 \).

- **Explanation**: \( \angle 7 \) and \( \angle 10 \) are alternate interior angles since the lines \( s \parallel t \) are parallel and a transversal crosses them. Alternate interior angles are congruent when lines are parallel. So:
  \[
  m\angle 10 = m\angle 7 = 84^\circ
  \]

### Problem 8
Given \( s \parallel t \), \( m\angle 8 = 8x - 4 \), and \( m\angle 16 = 6x + 24 \), find the value of \( x \).

- **Explanation**: \( \angle 8 \) and \( \angle 16 \) are corresponding angles, meaning they are equal when lines are parallel. Therefore:
  \[
  8x - 4 = 6x + 24
  \]
  Solving for \( x \):
  \[
  8x - 6x = 24 + 4
  \]
  \[
  2x = 28
  \]
  \[
  x = 14
  \]

### Problem 9
What do \( m\angle 2 \) and \( m\angle 3 \) need to do for \( a \parallel b \)?

- **Explanation**: For lines \( a \parallel b \), angles \( \angle 2 \) and \( \angle 3 \) are same-side interior angles. For \( a \parallel b \), these angles must be supplementary, meaning:
  \[
  m\angle 2 + m\angle 3 = 180^\circ
  \]

### Problem 10
If \( \angle 2 \) and \( \angle 4 \) are congruent, which two lines are parallel?

- **Explanation**: If \( \angle 2 \) and \( \angle 4 \) are congruent, they are alternate interior angles. Therefore, the lines \( t \parallel b \).

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Would you like further details or explanations on any of these problems? Here are some related questions to expand on this:

1. How do alternate interior angles behave when lines are not parallel?
2. What is the significance of corresponding angles in parallel lines?
3. Can you always use the same method for solving for \( x \) in angle equations?
4. How can you verify if two lines are parallel given a pair of angles?
5. What is the definition of supplementary angles?

**Tip**: Always remember that congruent alternate interior and corresponding angles are key indicators of parallel lines.

Given parallel lines s and t with multiple angles formed, determine specific angle measures, parallelism conditions, and solve for x.

This problem focuses on solving angles formed by parallel lines and a transversal, using key concepts like alternate interior angles, corresponding angles, and supplementary angles. Learn how to find angle measures and solve for x step by step.

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