The given problem involves parallel lines cut by a transversal. We are tasked to find the measures of angles $\angle 5$ and $\angle 7$ given that $\angle 4 = 121^\circ$.

Step-by-Step Solution:

1. Relationships Between Angles:

   • $\angle 4$ and $\angle 5$ are corresponding angles. For parallel lines cut by a transversal, corresponding angles are equal.
     Therefore, $m\angle 5 = m\angle 4 = 121^\circ$.

   • $\angle 4$ and $\angle 7$ are consecutive interior angles (same-side interior). These angles are supplementary, meaning their measures add up to $180^\circ$.
     So, $m\angle 4 + m\angle 7 = 180^\circ$.

2. Calculations:

   • $m\angle 5 = 121^\circ$ (from corresponding angles).
   • $m\angle 7 = 180^\circ - m\angle 4 = 180^\circ - 121^\circ = 59^\circ$.

Final Answer:

• $m\angle 5 = 121^\circ$
• $m\angle 7 = 59^\circ$

Would you like further explanation, or do you have other questions?

Related Questions:

1. What are the properties of angles formed by parallel lines and a transversal?
2. What is the relationship between alternate interior angles in this scenario?
3. How can you determine $\angle 6$ using the given information?
4. What happens if the lines are not parallel—how would this affect the angles?
5. How do you find angles in other geometric setups like triangles or polygons?

Tip:

Always label corresponding angles, alternate interior angles, and consecutive interior angles when working with parallel lines and a transversal to simplify problem-solving.