To find the missing angle measurements in the image, let's use the properties of parallel lines and transversals.

Step-by-Step Solution:

1. Identify Given Angles:

   • An angle marked as $63^\circ$ (let's call this $\angle A$).
   • An angle marked as $54^\circ$ (let's call this $\angle B$).

2. Properties of Angles Formed by a Transversal with Parallel Lines:

   • Corresponding Angles: Angles that are in the same position relative to the parallel lines and the transversal are equal.
   • Alternate Interior Angles: Angles that are on opposite sides of the transversal but inside the parallel lines are equal.
   • Supplementary Angles: Angles on the same line are supplementary, meaning their measures add up to $180^\circ$.

3. Analyzing $\angle A = 63^\circ$:

   • Since the lines are parallel, the corresponding angle on the opposite side of the transversal will also be $63^\circ$.
   • The angle that forms a straight line with $63^\circ$ (let's call this $\angle C$) is supplementary to $\angle A$. So, $\angle C = 180^\circ - 63^\circ = 117^\circ$.

4. Analyzing $\angle B = 54^\circ$:

   • The corresponding angle on the opposite side of the transversal will also be $54^\circ$.
   • The angle that forms a straight line with $54^\circ$ (let's call this $\angle D$) is supplementary to $\angle B$. So, $\angle D = 180^\circ - 54^\circ = 126^\circ$.

Summary of Missing Angle Measurements:

• Angles Corresponding to $63^\circ$ and $54^\circ$:

  • $\angle A = 63^\circ$
  • The corresponding angle to $\angle A$ across the transversal is also $63^\circ$.
  • $\angle B = 54^\circ$
  • The corresponding angle to $\angle B$ across the transversal is also $54^\circ$.

• Supplementary Angles:

  • $\angle C = 117^\circ$ (Supplementary to $63^\circ$)
  • $\angle D = 126^\circ$ (Supplementary to $54^\circ$)

Would you like further details or explanations on any part of this solution? Here are some related questions for further understanding:

1. What are alternate exterior angles, and how do they compare with alternate interior angles?
2. How would the angles change if the lines were not parallel?
3. How do we prove lines are parallel using angle relationships?
4. What are the key differences between corresponding, alternate interior, and alternate exterior angles?
5. How can we apply these angle properties to solve real-world problems involving parallel lines?

Tip: Understanding the relationships between angles formed by a transversal cutting through parallel lines is crucial for solving many geometric problems. Practice identifying these angles and their properties to strengthen your geometry skills.