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 $180^\circ$.