The problem asks for the measure of angle $ABE$, where the diagram shows two intersecting lines. Given that angle $ABC$ is marked as $40^\circ$ and line $AE$ extends through point $B$, we can apply some geometric rules to solve this:

• The lines appear to form a pair of intersecting straight lines at point $B$.
• Angle $ABE$ and angle $ABC$ are adjacent angles on a straight line (they form a linear pair), meaning their sum must equal $180^\circ$ since they lie on a straight line.

To calculate the measure of angle $ABE$: $\text{Angle ABE} = 180^\circ - 40^\circ = 140^\circ$

Thus, the measure of angle $ABE$ is $\boxed{140^\circ}$.

Let me know if you'd like more details or have any further questions!

Here are 5 related questions:

1. What is the sum of angles around a point?
2. How can you determine the measure of a vertical angle?
3. What is a linear pair of angles, and what properties do they have?
4. How would you solve for an unknown angle in a triangle?
5. How are adjacent angles related on a straight line?

Tip: For intersecting lines, vertical angles are always congruent, and adjacent angles form a linear pair!