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In the image, the problem states that ∠EFG and ∠GFH are a linear pair. This means their measures add up to 180° since linear pairs are supplementary angles. You are given:

- \( m\angle EFG = 5n + 24 \)
- \( m\angle GFH = 2n + 37 \)

Since they are a linear pair, we can set up the equation:

\[
(5n + 24) + (2n + 37) = 180
\]

### Step 1: Simplify the equation

\[
7n + 61 = 180
\]

### Step 2: Solve for \(n\)

\[
7n = 180 - 61
\]
\[
7n = 119
\]
\[
n = \frac{119}{7} = 17
\]

### Step 3: Find the measures of the angles

Now that we know \(n = 17\), we can substitute this back into the expressions for \( m\angle EFG \) and \( m\angle GFH \):

- \( m\angle EFG = 5n + 24 = 5(17) + 24 = 85 + 24 = 109^\circ \)
- \( m\angle GFH = 2n + 37 = 2(17) + 37 = 34 + 37 = 71^\circ \)

### Final Answer:
- \( m\angle EFG = 109^\circ \)
- \( m\angle GFH = 71^\circ \)

Would you like further clarification on this solution?

### Related Questions:
1. What are supplementary angles, and why do linear pairs always sum to 180°?
2. How would this problem change if the angles were complementary instead of supplementary?
3. Could you solve this if one of the angle expressions was quadratic, like \(5n^2 + 24\)?
4. What if both angles were given in terms of different variables, say \(m \angle EFG = 5x + 24\) and \(m \angle GFH = 2y + 37\)?
5. How does the concept of linear pairs extend to three or more angles in a polygon?

#### Tip:
When solving for angles in a pair, always check if they are complementary (sum to 90°) or supplementary (sum to 180°) to set up your equation correctly.

∠EFG and ∠GFH are a linear pair. m∠EFG = 5n + 24, and m∠GFH = 2n + 37. What are m∠EFG and m∠GFH?

Learn how to solve for m∠EFG and m∠GFH when given the expressions 5n + 24 and 2n + 37. This problem involves key algebraic and geometric concepts, including linear pair theorem and supplementary angles.

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