Question

quadrilateral qrst has vertices at q (-1,5), r (2,8), s (4,4), and t (1,1). the measure of angle q is 108° and the measure of angle r is 108 - 2x. in 3-5 sentences, determine if quadrilateral qrst is a parallelogram, and if so, solve for x. (4 points)

Explanation:

Step1: Calculate the slopes of sides

Slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
Slope of \(QR\): \(m_{QR}=\frac{8 - 5}{2-(-1)}=\frac{3}{3} = 1\)
Slope of \(TS\): \(m_{TS}=\frac{1 - 4}{1 - 4}=\frac{-3}{-3}=1\)
Slope of \(RS\): \(m_{RS}=\frac{4 - 8}{4 - 2}=\frac{-4}{2}=-2\)
Slope of \(TQ\): \(m_{TQ}=\frac{5 - 1}{-1 - 1}=\frac{4}{-2}=-2\)
Since \(m_{QR}=m_{TS}\) and \(m_{RS}=m_{TQ}\), opposite sides are parallel.

Step2: Use angle property of parallelogram

In a parallelogram, consecutive angles are supplementary. So \(\angle Q+\angle R = 180^{\circ}\)
Substitute \(\angle Q = 108^{\circ}\) and \(\angle R=108 - 2x\) into \(\angle Q+\angle R = 180^{\circ}\)
\(108+(108 - 2x)=180\)
\(216-2x = 180\)
\(-2x=180 - 216\)
\(-2x=-36\)
\(x = 18\)

Answer:

Quadrilateral \(QRST\) is a parallelogram and \(x = 18\)