This is a geometry problem involving angles in a right triangle. We are given two angles, $(5r + 7)^\circ$ and $(7r - 1)^\circ$, and asked to solve for $r$.

Step-by-Step Solution:

1. Recognizing the Angle Sum Property: In a right triangle, the sum of the angles must always be $90^\circ$ since one of the angles is a right angle (i.e., $90^\circ$).

   The two angles given in the diagram are:

   • $(5r + 7)^\circ$
   • $(7r - 1)^\circ$

   Therefore, we can set up the equation based on the angle sum property: $(5r + 7) + (7r - 1) = 90$

2. Solving the Equation: Let's simplify and solve for $r$: $5r + 7 + 7r - 1 = 90$ Combine like terms: $12r + 6 = 90$ Subtract 6 from both sides: $12r = 84$ Divide by 12: $r = 7$

Thus, the value of $r$ is $7$.

Would you like further explanation or additional details on any part of this solution? Here are some related questions that could deepen your understanding:

1. What is the angle sum property of a triangle?
2. How does solving linear equations help in geometry?
3. What are some other properties of angles in a triangle?
4. How would this problem change if we had a different type of triangle?
5. Can you solve for unknown angles when given the sine or cosine in a right triangle?

Tip: Remember, the sum of all angles in any triangle always adds up to 180 degrees!