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To solve this problem, we need to find the degree measures of each angle in the triangle.

### Given Information:
1. The triangle is a right triangle with \(\angle F = 90^\circ\).
2. The measures of the other two angles are given as:
   - \(m\angle G = (4x + 9)^\circ\)
   - \(m\angle H = (5x - 27)^\circ\)

### Steps to solve:
1. **Sum of angles in a triangle**: 
   The sum of the interior angles in any triangle is \(180^\circ\). Hence, we have the equation:
   \[
   m\angle F + m\angle G + m\angle H = 180^\circ
   \]
   Substituting the known values:
   \[
   90^\circ + (4x + 9)^\circ + (5x - 27)^\circ = 180^\circ
   \]

2. **Simplify the equation**:
   \[
   90 + 4x + 9 + 5x - 27 = 180
   \]
   Combine like terms:
   \[
   4x + 5x + 90 + 9 - 27 = 180
   \]
   \[
   9x + 72 = 180
   \]

3. **Solve for \(x\)**:
   Subtract 72 from both sides:
   \[
   9x = 108
   \]
   Divide by 9:
   \[
   x = 12
   \]

4. **Find the angle measures**:
   Now that we have \(x = 12\), we can substitute this value back into the expressions for \(m\angle G\) and \(m\angle H\):

   - \(m\angle G = 4x + 9 = 4(12) + 9 = 48 + 9 = 57^\circ\)
   - \(m\angle H = 5x - 27 = 5(12) - 27 = 60 - 27 = 33^\circ\)

### Final Answer:
- \(m\angle F = 90^\circ\)
- \(m\angle G = 57^\circ\)
- \(m\angle H = 33^\circ\)

Would you like more details on any of these steps, or help with another question?

### Related Questions:
1. How do we know the sum of angles in a triangle is always 180 degrees?
2. Can the value of \(x\) ever be negative in such problems?
3. What happens if we are given a different type of triangle (e.g., isosceles)?
4. How can we verify if the answers are correct?
5. Could the measures of \(\angle G\) and \(\angle H\) be swapped in this problem?

### Tip:
In right triangles, once you know two angles, you can always find the third by subtracting their sum from 180 degrees.

In the triangle below, with right angle ∠F, suppose that m∠G = (4x + 9)° and m∠H = (5x - 27)°. Find the degree measure of each angle in the triangle.

Learn how to find the measures of angles in a right triangle with expressions for angles G and H. Given ∠G = (4x + 9)° and ∠H = (5x - 27)°, find the value of x and the degree measures of each angle. Suitable for Grades 8-10.

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