Figure 5. Given △ROP, RO = OP, and RO ⊥ OP. What is the measure of ∠ORP?

Solution:

We are given a triangle △ROP, where RO = OP and RO ⊥ OP.

1. RO = OP: This means that △ROP is an isosceles triangle with base RP. In an isosceles triangle, the angles opposite the equal sides are equal. Therefore, ∠ORP = ∠OPR.
2. RO ⊥ OP: This means that the angle between RO and OP is 90 degrees. So, ∠ROP = 90°.
3. The sum of angles in any triangle is 180°. In △ROP, we have: ∠ROP + ∠ORP + ∠OPR = 180°.
4. Substitute the known values: 90° + ∠ORP + ∠OPR = 180°.
5. Since ∠ORP = ∠OPR, we can write: 90° + ∠ORP + ∠ORP = 180°.
6. Combine like terms: 90° + 2∠ORP = 180°.
7. Subtract 90° from both sides: 2∠ORP = 180° - 90°.
8. 2∠ORP = 90°.
9. Divide by 2: ∠ORP = \( \frac{90°}{2} \).
10. ∠ORP = 45°.

Answer: The measure of ∠ORP is 45°.