In triangle, E, F, G, comma△EFG, start overline, G, E, end overline, \cong, start overline, F, G, end overline 
GE
 ≅ 
FG
  and m, angle, G, equals, 120, degrees, .m∠G=120 
∘
 . Find m, angle, F, .m∠F

To solve this problem, let's analyze the given information step-by-step:

### Given:
1. Triangle \( \triangle EFG \)
2. \( \overline{GE} \cong \overline{FG} \), meaning sides \( GE \) and \( FG \) are congruent.
3. \( m\angle G = 120^\circ \)

### To Find:
The measure of \( m\angle F \).

### Solution:
Since \( \overline{GE} \cong \overline{FG} \), triangle \( \triangle EFG \) is **isosceles** with \( GE = FG \). In an isosceles triangle, the angles opposite the congruent sides are equal. Here, \( \angle F \) and \( \angle E \) are the base angles opposite sides \( GE \) and \( FG \), respectively. Thus, we have:

\[
m\angle F = m\angle E
\]

The sum of all interior angles in any triangle is \( 180^\circ \). So, we can write:

\[
m\angle E + m\angle F + m\angle G = 180^\circ
\]

Since \( m\angle F = m\angle E \), we can substitute to get:

\[
2m\angle F + 120^\circ = 180^\circ
\]

### Solving for \( m\angle F \):

\[
2m\angle F = 180^\circ - 120^\circ
\]

\[
2m\angle F = 60^\circ
\]

\[
m\angle F = \frac{60^\circ}{2}
\]

\[
m\angle F = 30^\circ
\]

### Conclusion:
The measure of \( m\angle F \) is \( 30^\circ \).

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

### Related Questions:
1. What is the measure of \( m\angle E \) in the same triangle \( \triangle EFG \)?
2. How do you prove that a triangle is isosceles using its angles?
3. Can a triangle have more than one angle measuring \( 120^\circ \)?
4. How do you find the area of an isosceles triangle if the base and height are known?
5. What are the properties of an equilateral triangle?

### Tip:
Remember that in any isosceles triangle, the angles opposite the congruent sides are always equal. This is a key property that can be used to solve many geometry problems involving isosceles triangles.

In triangle EFG, GE ≅ FG and m∠G = 120°. Find m∠F.

Discover how to find the measure of m∠F in an isosceles triangle where GE ≅ FG and m∠G = 120°. This geometry problem involves understanding the properties of isosceles triangles and the angle sum theorem. Suitable for Grades 8-10.

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