Question

given that the quadrilateral is a parallelogram,
$m\angle s = 2x + 17$ and $m\angle t = 6x - 5$, what is $m\angle q$?
$m\angle q =$______

Explanation:

Step1: Use the property of consecutive angles in a parallelogram

In a parallelogram, consecutive angles are supplementary. So, \(m\angle S+m\angle T = 180^{\circ}\).
Given \(m\angle S=2x + 17\) and \(m\angle T=6x-5\), we have the equation \((2x + 17)+(6x-5)=180\).
Simplify the left - hand side: \(2x+17 + 6x-5=8x + 12\).
So, \(8x+12 = 180\).
Subtract 12 from both sides: \(8x=180 - 12=168\).
Divide both sides by 8: \(x=\frac{168}{8}=21\).

Step2: Find \(m\angle S\)

Substitute \(x = 21\) into \(m\angle S=2x + 17\).
\(m\angle S=2\times21+17=42 + 17=59^{\circ}\).

Step3: Use the property of opposite angles in a parallelogram

In a parallelogram, \(\angle Q\) and \(\angle T\) are also consecutive angles (or we can use the property that \(\angle Q\) and \(\angle S\) are consecutive angles. Since \(\angle Q\) and \(\angle S\) are consecutive angles in a parallelogram, \(m\angle Q=180^{\circ}-m\angle S\).
Since \(m\angle S = 59^{\circ}\), then \(m\angle Q=180 - 59=121^{\circ}\).

Answer:

\(121^{\circ}\)