https://file.aiquickdraw.com/mathbot/file/1726111462113jmidzts7.jpg

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 \(7\).

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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!

Solve for the value of r. The two angles in a right triangle are (5r + 7)° and (7r - 1)°.

Learn how to solve for the value of r in a right triangle with angles (5r + 7)° and (7r - 1)°. This problem involves algebraic expressions and the angle sum property of triangles. Suitable for students in Grades 8-10.

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