We are given two angles:

• $m \angle a = 33^\circ$
• $m \angle b = 54^\circ$

We are tasked with finding the measures of angles $x$ and $y$. From the diagram, it is evident that:

1. $\angle x$ and $\angle a$ are vertically opposite angles.
2. $\angle y$ and $\angle b$ are also vertically opposite angles.

Step-by-step solution:

1. For $m \angle x$: Vertically opposite angles are always equal, so: $m \angle x = m \angle a = 33^\circ$

2. For $m \angle y$: Similarly, since $\angle y$ and $\angle b$ are vertically opposite angles: $m \angle y = m \angle b = 54^\circ$

Final answer:

• $m \angle x = 33^\circ$
• $m \angle y = 54^\circ$

Would you like more details on vertical angles or any other concept? Here are some follow-up questions that expand on this:

1. What are supplementary angles, and how could they apply here?
2. How do we identify corresponding angles in parallel line diagrams?
3. Can you find the relationship between interior and exterior angles in polygons?
4. How would the diagram change if the lines were not intersecting?
5. What is the sum of the interior angles around a point?

Tip: Vertically opposite angles are always congruent when two lines intersect, a key property in geometry!