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The given diagram appears to show a geometric figure involving right angles and congruent sides, which implies that there are some key properties we can use to solve for the measure of \( m \angle FGI \).

### Observations:
1. The triangles \( \triangle IFG \) and \( \triangle HIG \) appear to be right triangles since \( \angle IFH \) and \( \angle HGI \) are marked as right angles (90°).
2. The sides \( IF \) and \( IH \) are marked as congruent, which suggests that \( \triangle IFH \) is an isosceles triangle.
3. The angle \( \angle HGI \) is given in terms of \( a \): \( \angle HGI = \frac{a + 17^\circ}{2a} \), which relates to the unknown variable \( a \).

### Steps to solve for \( m \angle FGI \):

1. Since \( \triangle IFH \) is isosceles and \( \angle IFH = 90^\circ \), the other two angles, \( \angle FIH \) and \( \angle HFI \), are equal. This means \( \angle FIH = \angle HFI = 45^\circ \).
2. The angle \( \angle FGI \) can be found using the triangle sum property: the sum of the angles in any triangle is 180°.
   - In \( \triangle FGI \), we already know:
     - \( \angle HFI = 45^\circ \) (since it's part of the isosceles triangle),
     - \( \angle HGI = \frac{a + 17^\circ}{2a} \).

3. To find the value of \( m \angle FGI \), we need the specific value of \( a \), but without this information directly, we must assume that the question is asking us to express \( m \angle FGI \) in terms of \( a \).

Would you like to proceed with an algebraic solution or further explanation? Let me know if you need clarification or if you have any specific details to discuss.

### Relative Questions:
1. What are the properties of an isosceles right triangle?
2. How can we use the triangle sum property to solve for unknown angles?
3. How do congruent sides help determine angles in triangles?
4. What role does symmetry play in this type of geometric figure?
5. How can angles in terms of variables (like \( a \)) be solved in other geometry problems?

**Tip**: In any geometric figure, identifying congruent sides and right angles can simplify finding unknown angles.

What is the measure of angle FGI in the given geometric figure?

Learn how to calculate the measure of angle FGI in a geometric figure involving right angles and isosceles triangles. This problem explores the triangle sum property and angle relationships, suitable for students in Grades 9-10.

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