In ΔGHI,  start overline, G, I, end overline 
GI
  is extended through point I to point J,  m, angle, H, I, J, equals, left parenthesis, 9, x, minus, 15, right parenthesis, degreesm∠HIJ=(9x−15) 
∘
 ,  m, angle, I, G, H, equals, left parenthesis, 3, x, plus, 13, right parenthesis, degreesm∠IGH=(3x+13) 
∘
 ,   and m, angle, G, H, I, equals, left parenthesis, 3, x, plus, 8, right parenthesis, degreesm∠GHI=(3x+8) 
∘
 .   Find m, angle, I, G, H, .m∠IGH.

We are given a triangle \( \Delta GHI \), and the following angle measures:

- \( m\angle HIJ = (9x - 15)^\circ \)
- \( m\angle IGH = (3x + 13)^\circ \)
- \( m\angle GHI = (3x + 8)^\circ \)

Since \( \overline{GI} \) is extended to point \( J \), angle \( \angle HIJ \) becomes an **exterior angle** to the triangle. By the **exterior angle theorem**, an exterior angle is equal to the sum of the two non-adjacent interior angles of the triangle.

Thus,  
\[
m\angle HIJ = m\angle IGH + m\angle GHI
\]
Substitute the given values:
\[
9x - 15 = (3x + 13) + (3x + 8)
\]
Simplify the right side:
\[
9x - 15 = 6x + 21
\]
Now, solve for \( x \):
\[
9x - 6x = 21 + 15
\]
\[
3x = 36
\]
\[
x = 12
\]

### Step 2: Find \( m\angle IGH \)  
Now that we know \( x = 12 \), substitute it into the expression for \( m\angle IGH = 3x + 13 \):

\[
m\angle IGH = 3(12) + 13 = 36 + 13 = 49^\circ
\]

### Final Answer  
The measure of \( m\angle IGH \) is:

\[
\boxed{49^\circ}
\]

---

Would you like more details or have further questions?  

Here are some follow-up questions related to this problem:  
1. How do you prove the exterior angle theorem?  
2. What are other applications of the exterior angle theorem?  
3. Can the interior angles of any triangle exceed \(90^\circ\)?  
4. How do you find missing angles in non-right triangles?  
5. What is the sum of the interior angles of a triangle?  

**Tip:** Always check if a problem involves exterior angles—they often make finding missing angles easier!

In ΔGHI, GI is extended through point I to point J. m∠HIJ = (9x - 15)°, m∠IGH = (3x + 13)°, and m∠GHI = (3x + 8)°. Find m∠IGH.

In this problem, you will learn to apply the Exterior Angle Theorem to find m∠IGH in ΔGHI. The angles are given as algebraic expressions, and solving the problem requires finding the value of x. Suitable for Grades 8-10, this problem covers key geometry concepts and solving linear equations.

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