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To solve this problem, we need to find the value of \( m \angle HGF \).

### Information given:
- \( \angle HGF = 8x + 36 \)
- \( m \angle HJI = 3x + 66 \)
- \( I \) and \( J \) are midpoints of \( FH \) and \( GH \), respectively.

### Key observation:
Since \( I \) and \( J \) are midpoints of sides \( FH \) and \( GH \), segment \( IJ \) is parallel to \( FG \) by the Midline Theorem (or Midsegment Theorem). This means that \( \triangle HJI \) is similar to \( \triangle HGF \).

Thus, the corresponding angles are equal:
\[
m \angle HJI = m \angle HGF
\]

### Step-by-step solution:
1. Set \( m \angle HGF = m \angle HJI \) because of the similarity:
   \[
   8x + 36 = 3x + 66
   \]

2. Solve for \( x \):
   \[
   8x - 3x = 66 - 36
   \]
   \[
   5x = 30
   \]
   \[
   x = 6
   \]

3. Substitute \( x = 6 \) back into the expression for \( m \angle HGF \):
   \[
   m \angle HGF = 8(6) + 36 = 48 + 36 = 84^\circ
   \]

### Final Answer:
The measure of \( m \angle HGF \) is \( 84^\circ \).

Let me know if you'd like further details or clarifications. Here are some related questions for practice:

1. What is the Midline Theorem, and how does it apply to this problem?
2. What are the properties of similar triangles, and how do they help solve this problem?
3. How can we verify that \( \angle HJI \) corresponds to \( \angle HGF \)?
4. Can we apply this method to non-triangular shapes?
5. How do midpoints affect the angles and side lengths of a triangle?

**Tip:** In geometry problems involving midpoints, always consider the Midline Theorem, which can simplify your calculations by relating smaller triangles to the original shape.

In the diagram of triangle FGH, I is the midpoint of FH and J is the midpoint of GH. If m∠HGF = 8x + 36, and m∠HJI = 3x + 66, what is the measure of ∠HGF?

In this problem, you will learn how to solve for the angle ∠HGF in triangle FGH using the midline theorem and similar triangles. The problem involves algebraic equations and key geometric concepts suitable for Grades 9-10 students.

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