To solve the problem, we’ll go through each part step by step.
Part A: Finding the value of x
Given:
- \( m\angle OPQ = (x + 17)^\circ \)
- \( m\angle RPS = (8x - 8)^\circ \)
Since angles OPQ and RPS are complementary, this means that their measures add up to \(90^\circ\):
\[ m\angle OPQ + m\angle RPS = 90^\circ \]
Substituting in the expressions for the angles, we have:
\[ (x + 17) + (8x - 8) = 90 \]
Now combine like terms:
\[ x + 8x + 17 - 8 = 90 \] \[ 9x + 9 = 90 \]
Next, subtract \(9\) from both sides:
\[ 9x = 90 - 9 \] \[ 9x = 81 \]
Now, divide by \(9\):
\[ x = \frac{81}{9} = 9 \]
So, the value of \(x\) is \(9\).
Part B: Finding the measures of angles OPQ and RPS
Using the value of \(x\) from Part A, we can now find the measures of the angles:
-
Angle OPQ: \[ m\angle OPQ = x + 17 \] \[ m\angle OPQ = 9 + 17 = 26^\circ \]
-
Angle RPS: \[ m\angle RPS = 8x - 8 \] \[ m\angle RPS = 8(9) - 8 = 72 - 8 = 64^\circ \]
Thus, the measures of the angles are:
- \( m\angle OPQ = 26^\circ \)
- \( m\angle RPS = 64^\circ \)
Part C: Could the angles also be vertical angles?
Vertical angles are angles that are opposite each other when two lines intersect. In this situation, however, angles OPQ and RPS are defined as complementary angles not vertical angles.
- Since OPQ and RPS sum up to \(90^\circ\), they cannot be vertical angles because vertical angles must be equal to one another and should sum up to \(180^\circ\).
Therefore, angles OPQ and RPS cannot be vertical angles as vertical angles are formed in the context of two intersecting lines and do not usually sum to \(90^\circ\).
In conclusion:
- \( x = 9 \)
- \( m\angle OPQ = 26^\circ \), \( m\angle RPS = 64^\circ \)
- No, angles OPQ and RPS cannot be vertical angles because they are complementary.