https://file.aiquickdraw.com/mathbot/file/1725459469913j3fsxi3g.jpg

To solve the problem, let's analyze the given information from the image:

### Information Given:
- The front of an A-frame house is in the shape of an isosceles triangle.
- \( HK \cong GJ \), \( GH \cong JH \), and \( m\angle G = 72.5^\circ \).
- We need to determine the measure of \( m\angle GHK \).

### Steps to Solve:

1. **Recognize the isosceles triangle:**
   - Since \( HK \cong GJ \) and \( GH \cong JH \), \( \triangle GHK \) and \( \triangle JHK \) are isosceles triangles. 

2. **Calculate the base angles of \( \triangle GHJ \):**
   - \( \triangle GHJ \) is also isosceles with \( GH \cong JH \). Therefore, \( \angle G = \angle J = 72.5^\circ \).
   - The angle \( \angle GHJ \) is the vertex angle of the isosceles triangle \( \triangle GHJ \).

3. **Determine the vertex angle \( \angle GHJ \):**
   - The sum of the angles in any triangle is \( 180^\circ \).
   - Since \( \angle G = \angle J = 72.5^\circ \):
   \[
   \angle GHJ = 180^\circ - (72.5^\circ + 72.5^\circ) = 180^\circ - 145^\circ = 35^\circ
   \]

4. **Relate to \( \triangle GHK \):**
   - Since \( \triangle GHK \) is isosceles and shares the same vertex \( H \) with \( \triangle GHJ \), \( m\angle GHK \) should be calculated considering that \( \angle GHJ = \angle GHK \).

Thus, the measure of \( m\angle GHK \) is \( 35^\circ \).

### Answer:
\[ m\angle GHK = 35^\circ \]

Would you like a detailed explanation of any of the steps above or further clarification?

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### Relative Questions:
1. What properties define an isosceles triangle?
2. How do you calculate the angles of a triangle when given one angle and the properties of the triangle?
3. What is the significance of congruent sides in determining the angles of a triangle?
4. How can you determine if a triangle is isosceles based on its angle measures?
5. What are other methods to verify the angle in \( \triangle GHK \)?

### Tip:
Always remember that the sum of angles in any triangle is \( 180^\circ \). This fact is fundamental in solving various geometric problems.

The front of an A-frame house is in the shape of an isosceles triangle, as shown in the diagram. In the diagram, HK ≅ GJ, GH ≅ JH, and ∠G = 72.5°. Determine the m∠GHK.

This problem involves solving an isosceles triangle to find the angle m∠GHK. Given that ∠G = 72.5°, HK ≅ GJ, and GH ≅ JH, students will use the sum of angles in a triangle and the Isosceles Triangle Theorem. Suitable for Grades 8-10.

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