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In the given image, you are asked to determine the measure of \( \angle ABC \). Let's analyze the diagram step by step:

1. It is given that \( \overline{AD} \perp \overline{BE} \), meaning line \( AD \) is perpendicular to line \( BE \), forming a right angle at point \( B \). This tells us that \( \angle ABE \) is 90°.
2. Two angles are labeled in the diagram: 
   - \( \angle ABC = (60 - x)^\circ \),
   - \( \angle FBC = (2x + 5)^\circ \).

Since both of these angles share the same vertex \( B \) and lie along the straight line \( BE \), they are supplementary, meaning they must add up to 180°.

So, we can set up the following equation:
\[
(60 - x) + (2x + 5) = 180
\]

Simplifying this:
\[
60 - x + 2x + 5 = 180
\]
\[
x + 65 = 180
\]
\[
x = 115
\]

Now that we know \( x = 115 \), we can find the measure of \( \angle ABC \):
\[
\angle ABC = (60 - x) = 60 - 115 = -55^\circ
\]

There seems to be an inconsistency, as a negative angle is not physically meaningful in this case. I will need to recheck the interpretation of the diagram or conditions. Could you confirm any additional constraints, or I will try another approach with a different assumption?

Let me know if you'd like more clarification or further details!



Determine the measure of \( \angle ABC \) given that \( \overline{AD} \perp \overline{BE} \) and \( \angle ABC = (60 - x)^\circ \), \( \angle FBC = (2x + 5)^\circ \).

Learn how to determine the measure of \( \angle ABC \) in a geometry problem involving perpendicular lines and supplementary angles. This problem covers key algebraic and geometric concepts, including angle relationships and supplementary angles. Suitable for students in Grades 8-10.

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