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In this problem, we are given the following:

- \( EH \cong GH \), meaning that line segments \( EH \) and \( GH \) are congruent (i.e., equal in length).
- \( m\angle EFH = 45^\circ \), which is the measure of angle \( EFH \).
- We are asked to find \( m\angle GFH \), which is the measure of angle \( GFH \).

### Steps to solve:
1. The quadrilateral \( HGEF \) forms a kite, where two pairs of adjacent sides are congruent. \( EH \cong GH \), and the angles formed at point \( H \) involve right angles.
2. Since \( \angle EFH \) is given as \( 45^\circ \) and the angles at \( E \) and \( G \) are both right angles, \( \angle EHF \) and \( \angle GHF \) form complementary angles.
3. Using the fact that the sum of angles in any triangle is \( 180^\circ \), and angle \( EFH \) is already \( 45^\circ \), we can find \( \angle GFH \) by applying the angle sum property.

### Angle Calculation:
Since \( \angle EFH = 45^\circ \), we can say:
\[
m\angle GFH = 90^\circ - 45^\circ = 45^\circ
\]

Thus, the measure of angle \( GFH \) is **45°**.

Let me know if you'd like a more detailed explanation or have further questions!

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### Follow-up questions:
1. How would the solution change if \( EH \) were not congruent to \( GH \)?
2. Can you explain why the right angles play a role in this solution?
3. What is the significance of the given \( m\angle EFH = 45^\circ \)?
4. How would you calculate \( m\angle EGH \) if needed?
5. Could you solve this problem using trigonometric ratios?

### Tip:
Always check for congruent sides or angles in geometric figures—they can simplify your calculations!

If EH ≅ GH and m∠EFH = 45°, what is m∠GFH?

This geometry problem involves finding the measure of angle GFH given that two sides of a kite are congruent and another angle is 45°. The problem requires using the angle sum property and recognizing congruent sides in a kite. Suitable for students in Grades 7-9.

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