To solve this problem, we need to understand the relationship between angles formed by two parallel lines cut by a transversal.

Given:

• Two parallel lines are cut by a transversal.
• $m \angle 4 = 102^\circ$.

We need to find:

• $m \angle 5$
• $m \angle 7$

Solution:

1. Identify Angle Relationships:

   • Angles on opposite sides of the transversal but inside the parallel lines (like angles 4 and 5) are called alternate interior angles.
   • Angles on the same side of the transversal and same position relative to the parallel lines (like angles 4 and 7) are called corresponding angles.

2. Calculate $m \angle 5$:

   • Since angles 4 and 5 are alternate interior angles, they are congruent.
   • Therefore, $m \angle 5 = m \angle 4 = 102^\circ$.

3. Calculate $m \angle 7$:

   • Since angles 4 and 7 are corresponding angles, they are also congruent.
   • Thus, $m \angle 7 = m \angle 4 = 102^\circ$.

Answer:

• $m \angle 5 = 102^\circ$
• $m \angle 7 = 102^\circ$

Would you like further details on the solution, or do you have any questions?

Here are 5 related questions to deepen your understanding:

1. What are alternate interior angles, and why are they equal when two lines are parallel?
2. How would you find the measure of angle 6 in this diagram?
3. What would change if the lines were not parallel?
4. How can we use corresponding angles to find unknown angles in similar problems?
5. Are there any scenarios where corresponding angles would not be equal?

Tip: Remember that when two parallel lines are cut by a transversal, alternate interior angles and corresponding angles are always congruent.