The question asks you to find the measure of each numbered angle in the figure, where lines $m$ and $n$ are parallel.

From the image, it appears that angles 1, 5, 7, and 10 are labeled, and other angles (such as 77° and 121°) are already given.

To solve this problem:

1. Corresponding Angles: When two parallel lines are crossed by a transversal, the angles in matching corners (corresponding angles) are equal.

   • Angle 5 corresponds to the given 77° (so $\text{m} \angle 5 = 77^\circ$).

2. Vertical Angles: Opposite angles formed by the intersection of two lines are congruent.

   • Angle 1 is vertically opposite angle 5, so $\text{m} \angle 1 = 77^\circ$.

3. Supplementary Angles: Angles on a straight line add up to 180°.

   • Angle 7 and the 121° form a linear pair, so $\text{m} \angle 7 = 180^\circ - 121^\circ = 59^\circ$.

4. Vertical Angles Again: Angle 10 is vertically opposite angle 7, so $\text{m} \angle 10 = 59^\circ$.

Summary of Angles:

• $\text{m} \angle 1 = 77^\circ$
• $\text{m} \angle 5 = 77^\circ$
• $\text{m} \angle 7 = 59^\circ$
• $\text{m} \angle 10 = 59^\circ$

Would you like more details or have any questions?

Here are five related questions you might explore:

1. What are alternate interior angles, and how do they relate to parallel lines?
2. How can you determine the measures of angles when more than one transversal crosses parallel lines?
3. How do supplementary angles work with non-parallel lines?
4. What are the key differences between corresponding and alternate angles?
5. How do you calculate exterior angles in polygons when parallel lines are involved?

Tip: Always look for relationships like corresponding or vertical angles when dealing with parallel lines crossed by a transversal.