https://file.aiquickdraw.com/mathbot/file/1726627799357mlj9r3o8.jpg

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!

Learn how to find the measures of m∠EBC and m∠ABE by applying the Vertically Opposite Angles Theorem and the Supplementary Angles Theorem. This geometry problem is suitable for Grades 6-8.

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