Question

given that the quadrilateral is a parallelogram,
$m\angle s=x + 20$ and $m\angle t=7x - 8$, what is $m\angle q$?
$139^{\circ}$
$150^{\circ}$
$154^{\circ}$
$146^{\circ}$
none of these answers are correct

Explanation:

Step1: Use the property of adjacent angles in a parallelogram

In a parallelogram, adjacent angles are supplementary, so \(m\angle S+m\angle T = 180^{\circ}\).
Given \(m\angle S=x + 20\) and \(m\angle T=7x-8\), we have the equation \((x + 20)+(7x-8)=180\).
Simplify the left - hand side: \(x+20 + 7x-8=8x + 12\).
So, \(8x+12 = 180\).

Step2: Solve the equation for \(x\)

Subtract 12 from both sides: \(8x=180 - 12=168\).
Divide both sides by 8: \(x=\frac{168}{8}=21\).

Step3: Find \(m\angle S\)

Substitute \(x = 21\) into \(m\angle S=x + 20\), then \(m\angle S=21+20 = 41^{\circ}\).

Step4: Use the property of opposite angles in a parallelogram

In a parallelogram, \(\angle Q\) and \(\angle T\) are supplementary (also, \(\angle Q\) and \(\angle S\) are adjacent, and adjacent angles in a parallelogram are supplementary).
Since \(m\angle S = 41^{\circ}\), and \(m\angle Q+m\angle S=180^{\circ}\) (adjacent angles of a parallelogram are supplementary).

Answer:

\(m\angle Q=180^{\circ}-41^{\circ}=139^{\circ}\), so the answer is \(139^{\circ}\) (the first option).