In this diagram, we have two intersecting lines, $l_1$ and $l_2$, with several angles marked at their intersection points. The given angle is $\angle 3 = 77^\circ$. Let's use the relationships between angles to find the measures of the other angles.

Step-by-step solution:

1. Vertical Angles: Angles that are opposite each other when two lines cross are called vertical angles, and they are always equal. This gives:

   • $\angle 3 = \angle 1 = 77^\circ$.

2. Linear Pair: Angles that form a straight line sum to $180^\circ$. This gives:

   • $\angle 3 + \angle 2 = 180^\circ$, so $\angle 2 = 180^\circ - 77^\circ = 103^\circ$.
   • $\angle 1 + \angle 4 = 180^\circ$, so $\angle 4 = 180^\circ - 77^\circ = 103^\circ$.

3. Corresponding Angles: When two parallel lines ($l_1$ and $l_2$) are cut by a transversal, corresponding angles are congruent. This gives:

   • $\angle 3 = \angle 5 = 77^\circ$ (corresponding angles).
   • $\angle 2 = \angle 6 = 103^\circ$.
   • $\angle 4 = \angle 8 = 103^\circ$.
   • $\angle 1 = \angle 7 = 77^\circ$.

Final answers:

• $\angle 1 = 77^\circ$
• $\angle 2 = 103^\circ$
• $\angle 3 = 77^\circ$
• $\angle 4 = 103^\circ$
• $\angle 5 = 77^\circ$
• $\angle 6 = 103^\circ$
• $\angle 7 = 77^\circ$
• $\angle 8 = 103^\circ$

Let me know if you'd like a more detailed explanation on any of these steps!

Here are 5 related questions you might find interesting:

1. How are vertical angles identified in any given diagram?
2. What is the relationship between alternate interior angles?
3. Can you explain how corresponding angles work with parallel lines?
4. How do linear pairs of angles sum up, and why?
5. What properties do angles formed by a transversal cutting through parallel lines have?

Tip:

When working with angles formed by parallel lines and a transversal, always look for angle pairs such as corresponding, alternate interior, and alternate exterior angles, as they help in quickly identifying equal angle measures.