Question

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)

Explanation:

Part A:

Step1: Use the property of complementary angles

Complementary angles sum to \(90^{\circ}\). So, \((x + 17)+(8x - 8)=90\).

Step2: Simplify the left - hand side

Combine like terms: \(x+8x+17 - 8=90\), which gives \(9x + 9=90\).

Step3: Solve for \(x\)

Subtract 9 from both sides: \(9x=90 - 9=81\). Then divide by 9: \(x=\frac{81}{9}=9\).

Part B:

Step1: Substitute \(x = 9\) into \(\angle OPQ\)

\(m\angle OPQ=x + 17\). Substituting \(x = 9\), we get \(m\angle OPQ=9+17 = 26^{\circ}\).

Step2: Substitute \(x = 9\) into \(\angle RPS\)

\(m\angle RPS=8x - 8\). Substituting \(x = 9\), we get \(m\angle RPS=8\times9-8=72 - 8=64^{\circ}\).

Part C:

Step1: Recall the definition of vertical angles

Vertical angles are equal. Since \(m\angle OPQ = 26^{\circ}\) and \(m\angle RPS=64^{\circ}\), and \(26
eq64\).

Answer:

Part A:

\(x = 9\)

Part B:

\(m\angle OPQ = 26^{\circ}\), \(m\angle RPS=64^{\circ}\)

Part C:

No, because vertical angles are equal. Here \(m\angle OPQ = 26^{\circ}\) and \(m\angle RPS = 64^{\circ}\), and \(26
eq64\).