https://file.aiquickdraw.com/mathbot/file/1730308262665pxrcsimq.jpg

To find the measure of \( \angle DEF \) in this diagram, we can use the information provided.

1. The figure shows that \( \angle DEF \) and \( \angle DFE \) are a pair of **linear angles**, meaning they form a straight line.
2. Since they form a straight line, the sum of \( \angle DEF \) and \( \angle DFE \) is \( 180^\circ \).

### Given:
- \( m \angle DEF = (8x - 34)^\circ \)
- \( m \angle DFE = (5x + 2)^\circ \)

Since \( \angle DEF \) and \( \angle DFE \) are supplementary:
\[
(8x - 34) + (5x + 2) = 180
\]

### Solving for \( x \):
1. Combine like terms:
   \[
   13x - 32 = 180
   \]

2. Add 32 to both sides:
   \[
   13x = 212
   \]

3. Divide by 13:
   \[
   x = 16
   \]

### Finding \( m \angle DEF \):
Now, substitute \( x = 16 \) back into \( m \angle DEF = (8x - 34) \):
\[
m \angle DEF = 8(16) - 34 = 128 - 34 = 94^\circ
\]

### Answer:
The measure of \( \angle DEF \) is \( 94^\circ \).

Would you like further details or have questions?

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#### Related Questions
1. What is the measure of \( \angle DFE \) after finding \( x \)?
2. How do you verify if two angles are supplementary?
3. Can you solve similar problems with other types of angle relationships (e.g., complementary)?
4. What is the importance of linear pairs in geometry?
5. How would the solution change if \( \angle DEF \) and \( \angle DFE \) were complementary instead?

#### Tip:
In geometry, always look for angle relationships (like supplementary or complementary) to simplify your calculations.

Given the figure shown, find \( m \angle DEF \).

Learn how to find the measure of \( \angle DEF \) in a geometry problem involving supplementary angles and linear pairs. This problem covers key concepts like algebra and supplementary angles, suitable for students in Grades 8-10.

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