Question

(Angle Relationships HC)

Angles OPQ and RPS have the following measures:

m∠OPQ = (x + 17)°, m∠RPS = (8x − 8)°

Part A: If angle OPQ and angle RPS are complementary angles, find the value of x. Show every step of your work. (4 points)

Part B: Use the value of x from Part A to find the measures of angles OPQ and RPS. Show every step of your work. (4 points)

Part C: Could the angles also be vertical angles? Explain. (4 points)

Answer 1

Part A: For complementary angles, 9x + 9 = 90, yielding x = 9. Part B: With x = 9, m∠OPQ = 26° and m∠RPS = 64°. Part C: No, as m∠OPQ and m∠RPS are not equal, violating the condition for vertical angles.

Part A: Finding the Value of x (Complementary Angles)

Given that angles OPQ and RPS are complementary, the sum of their measures is 90°:

m∠OPQ + m∠RPS = 90°

(x + 17) + (8x - 8) = 90

Combine like terms:

9x + 9 = 90

Subtract 9 from both sides:

9x = 81

Divide by 9:

x = 9

Part B: Finding the Measures of Angles OPQ and RPS

Substitute the value of x back into the angle measures:

m∠OPQ = (x + 17) = 9 + 17 = 26°

m∠RPS = (8x - 8) = (8 * 9) - 8 = 64°

part C: Could the Angles Also be Vertical Angles?

Vertical angles are equal, but m∠OPQ and m∠RPS are not equal in this case (26° ≠ 64°). Therefore, angles OPQ and RPS are not vertical angles.