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The problem asks for the measure of angle \( \angle DEF \).

From the diagram:
- The angle \( \angle DEF \) is expressed as \( (5x + 2)^\circ \).
- The adjacent angle \( \angle DFE \) is given as \( (8x - 34)^\circ \).

Since \( \angle DEF \) and \( \angle DFE \) are on a straight line, they are supplementary angles. Supplementary angles add up to \( 180^\circ \). Therefore, we can write the equation:

\[
(5x + 2) + (8x - 34) = 180
\]

Now, let's solve for \( x \):

\[
5x + 8x + 2 - 34 = 180
\]

\[
13x - 32 = 180
\]

\[
13x = 212
\]

\[
x = \frac{212}{13}
\]

\[
x = 16.31
\]

Now, substitute this value of \( x \) back into the expression for \( \angle DEF \):

\[
\angle DEF = 5x + 2 = 5(16.31) + 2 = 81.55 + 2 = 83.55^\circ
\]

Thus, the measure of \( \angle DEF \) is approximately \( 83.55^\circ \).

Would you like more details or clarification on this?

### Related Questions:
1. What are supplementary and complementary angles?
2. How do you verify if two angles are supplementary?
3. Can two angles be adjacent and not supplementary?
4. What if the angles were vertically opposite instead of adjacent?
5. How would the problem change if \( \angle DEF \) was on a different line segment?

### Tip:
When dealing with linear pairs, always remember they add up to \( 180^\circ \) because they form a straight line.

Learn how to find the measure of angle ∠DEF using the concept of supplementary angles and linear equations. This problem demonstrates how to set up and solve for x in a system of linear equations, suitable for students in Grades 8-10.

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