The problem involves finding the measures of angles $m \angle 1$ and $m \angle 3$, given that $m \angle 8 = 147^\circ$.

Solution:

Key Concepts:

1. Alternate Interior Angles: These are equal when two parallel lines are cut by a transversal.
2. Linear Pair: Adjacent angles on the same line add up to $180^\circ$.

Steps:

1. Find $m \angle 3$:

   • $\angle 8$ and $\angle 3$ are alternate interior angles.
   • Since $\angle 8 = 147^\circ$, $\angle 3 = 147^\circ$.

2. Find $m \angle 1$:

   • $\angle 1$ and $\angle 3$ form a linear pair, so their sum is $180^\circ$.
   • $\angle 1 = 180^\circ - 147^\circ = 33^\circ$.

Final Answer:

• $m \angle 1 = 33^\circ$
• $m \angle 3 = 147^\circ$

Would you like me to clarify or expand on any part of this explanation?

Here are 5 related questions to deepen your understanding:

1. What is the relationship between corresponding angles in parallel lines?
2. How do we determine the measure of vertical angles in such configurations?
3. Can you identify other pairs of congruent angles in the diagram?
4. What happens to angle measures if the parallel lines are no longer parallel?
5. How can the concept of supplementary angles be used to verify these results?

Tip: Always look for relationships like "alternate interior," "corresponding," and "linear pair" when working with parallel lines and transversals.