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)

Answer 1

Compute slopes: QR = (8−5)/(2−(−1)) = 1, ST = (1−4)/(1−4) = 1, RS = (4−8)/(4−2) = −2, QT = (1−5)/(1−(−1)) = −2. Since QR ∥ ST and RS ∥ QT, QRST is a parallelogram. Adjacent angles in a parallelogram are supplementary, so 108° + (108° − 2x) = 180°. Solving gives 216 − 2x = 180 → 2x = 36 → x = 18.