https://file.aiquickdraw.com/mathbot/file/1730383920767186jeuyv.jpg

To solve for \( m \angle GEF \), let's analyze the information provided and use the properties of triangles and angles.

Given:
- In \( \triangle EFG \), line \( EG \) is extended to point \( H \).
- \( m \angle EFG = (3x + 15)^\circ \)
- \( m \angle GEF = (2x + 6)^\circ \)
- \( m \angle FGH = (9x - 15)^\circ \)

Since \( EG \) is extended to \( H \), \( \angle FGH \) is an **exterior angle** of \( \triangle EFG \). According to the exterior angle theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent (or remote) interior angles.

Therefore,
\[
m \angle FGH = m \angle EFG + m \angle GEF
\]

### Step 1: Set Up the Equation
Substitute the given expressions:
\[
9x - 15 = (3x + 15) + (2x + 6)
\]

### Step 2: Simplify the Equation
Combine like terms on the right side:
\[
9x - 15 = 5x + 21
\]

### Step 3: Solve for \( x \)
Subtract \( 5x \) from both sides:
\[
4x - 15 = 21
\]

Add 15 to both sides:
\[
4x = 36
\]

Divide by 4:
\[
x = 9
\]

### Step 4: Substitute \( x = 9 \) to Find \( m \angle GEF \)
Now, substitute \( x = 9 \) into \( m \angle GEF = (2x + 6)^\circ \):
\[
m \angle GEF = 2(9) + 6 = 18 + 6 = 24^\circ
\]

### Final Answer
\[
m \angle GEF = 24^\circ
\]

Would you like more details on any step, or do you have any questions?

---

Here are 5 related questions you might explore:

1. What is the measure of \( m \angle EFG \) given \( x = 9 \)?
2. How can we use the exterior angle theorem in other geometric contexts?
3. What happens if we extend a different side of \( \triangle EFG \)?
4. How do we verify the sum of the interior angles of a triangle?
5. Can the value of \( x \) be negative in this scenario? Why or why not?

**Tip:** When dealing with exterior angles in triangles, remember that the exterior angle theorem only applies to angles not adjacent to the angle being extended.

In ΔEFG, EG is extended through point G to point H, m∠EFG = (3x + 15)°, m∠GEF = (2x + 6)°, and m∠FGH = (9x - 15)°. Find m∠GEF.

Learn how to find m∠GEF in ΔEFG where EG is extended to point H. This problem involves using the Exterior Angle Theorem and algebraic equations to solve for unknown angles. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation