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This problem involves a right triangle with angles labeled as \((7x + 5)^\circ\), \((3x - 5)^\circ\), and the right angle \((90^\circ)\). Since the angles in any triangle add up to \(180^\circ\), we can set up an equation to solve for \(x\).

### Step-by-Step Solution

1. **Set up the equation:**
   \[
   (7x + 5) + (3x - 5) + 90 = 180
   \]

2. **Combine like terms:**
   \[
   10x + 90 = 180
   \]

3. **Isolate the \(x\)-term:**
   \[
   10x = 90
   \]

4. **Solve for \(x\):**
   \[
   x = 9
   \]

5. **Substitute \(x = 9\) back to find the angle measures:**
   - For \((7x + 5)^\circ\):
     \[
     7(9) + 5 = 63 + 5 = 68^\circ
     \]
   - For \((3x - 5)^\circ\):
     \[
     3(9) - 5 = 27 - 5 = 22^\circ
     \]

6. **Verification:**
   - Check if the angles add up to \(180^\circ\):
     \[
     68^\circ + 22^\circ + 90^\circ = 180^\circ
     \]
   - The values are correct.

### Answer
The angles are:
- \((7x + 5)^\circ = 68^\circ\)
- \((3x - 5)^\circ = 22^\circ\)
- Right angle = \(90^\circ\)

Would you like a detailed breakdown of any specific steps or have further questions?

Here are 5 related questions you might find useful:
1. How can you use angle sums to solve for unknowns in any triangle?
2. How would the approach change if the triangle was not a right triangle?
3. How does the value of \(x\) affect each angle in this problem?
4. What other properties of right triangles could help solve problems like this?
5. Can you determine if a triangle is right just by using the angle measures?

**Tip:** Remember that the sum of the angles in any triangle always equals \(180^\circ\).

Given a right triangle with angles labeled as (7x + 5)°, (3x - 5)°, and the right angle (90°). Find the values of x and each angle.

Learn how to find unknown angle measures in a right triangle labeled with algebraic expressions. This problem involves setting up an equation using the sum of angles and solving for x. Suitable for students in Grades 8-10.

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