In the image you provided, there is an intersection of two lines, and one of the angles formed is labeled with $54^\circ$ at point $A$. The task asks to find the measures of two angles:

1. $\angle EBC$
2. $\angle ABE$

Solution:

1. $\angle EBC$:

Since the given angle $\angle DAB = 54^\circ$ and $\angle EBC$ is a vertically opposite angle, the two angles are congruent (i.e., equal). Thus: $m\angle EBC = 54^\circ$

2. $\angle ABE$:

Now, $\angle ABE$ is a straight angle formed by $\overrightarrow{AB}$ and $\overrightarrow{BE}$, which sum up to $180^\circ$. Since one part of this angle is $\angle DAB = 54^\circ$, the measure of $\angle ABE$ can be found by subtracting $54^\circ$ from $180^\circ$: $m\angle ABE = 180^\circ - 54^\circ = 126^\circ$

Final Answers:

1. $m\angle EBC = 54^\circ$
2. $m\angle ABE = 126^\circ$

Would you like further clarification or any details on these calculations?

Related Questions:

1. How are vertically opposite angles related?
2. What is the sum of angles on a straight line?
3. Can you explain alternate interior angles?
4. How do you determine supplementary angles?
5. Why are adjacent angles on a straight line supplementary?

Tip:

Vertically opposite angles are always equal when two lines intersect!