To determine the missing angle $\angle r$ in the triangle:

Step 1: Understand the problem

• The given triangle has:
  • One marked angle as $65^\circ$,
  • Two sides marked as equal, making it an isosceles triangle.
• In an isosceles triangle, the two base angles are congruent.

This means: $\angle r = \angle \text{other base angle} = x$

Step 2: Apply the triangle sum theorem

The sum of all angles in a triangle equals $180^\circ$. Let the two base angles both be $x$, and the vertex angle be $65^\circ$. Thus: $x + x + 65^\circ = 180^\circ$

Step 3: Solve for $x$

Combine like terms: $2x + 65^\circ = 180^\circ$

Subtract $65^\circ$ from both sides: $2x = 115^\circ$

Divide by 2: $x = 57.5^\circ$

Step 4: Conclusion

The missing base angle $\angle r$ is: $\boxed{57.5^\circ}$

Would you like a detailed explanation of any step? Here are some related questions to explore:

1. What is the triangle sum theorem?
2. How do you identify an isosceles triangle?
3. What is the property of base angles in an isosceles triangle?
4. Can you prove that base angles in an isosceles triangle are equal?
5. How do you calculate missing angles in a right triangle?

Tip: Always check if a triangle is isosceles, equilateral, or scalene to simplify angle calculations.