We are given that two parallel lines are cut by a transversal, and we know that $m\angle 8 = 97^\circ$. Our task is to find $m\angle 2$ and $m\angle 3$.

Step-by-Step Solution:

1. Relationship between $\angle 8$ and $\angle 2$:

   • Angles $\angle 2$ and $\angle 8$ are corresponding angles because they are on the same side of the transversal and the same side of the parallel lines.
   • For parallel lines, corresponding angles are congruent, so: $m\angle 2 = m\angle 8 = 97^\circ$

2. Relationship between $\angle 8$ and $\angle 3$:

   • Angles $\angle 8$ and $\angle 3$ are consecutive interior angles, or sometimes called same-side interior angles.
   • Consecutive interior angles are supplementary, meaning their measures add up to $180^\circ$. So: $m\angle 8 + m\angle 3 = 180^\circ$ Since $m\angle 8 = 97^\circ$, we have: $97^\circ + m\angle 3 = 180^\circ$ Solving for $m\angle 3$: $m\angle 3 = 180^\circ - 97^\circ = 83^\circ$

Final Answers:

• $m\angle 2 = 97^\circ$
• $m\angle 3 = 83^\circ$

Would you like further details or clarifications?

Here are 5 related questions you can explore:

1. What are alternate interior angles, and how do they relate to parallel lines?
2. If $m\angle 1 = 97^\circ$, what is $m\angle 4$?
3. How can you prove that two lines are parallel using angle relationships?
4. What are vertical angles, and how do they relate to this diagram?
5. How do transversal angles behave when the lines are not parallel?

Tip: Always remember, corresponding angles are congruent when two parallel lines are cut by a transversal!