Question

Quadrilateral QRST has vertices at Q(−1,5)%0D%0A𝑄%0D%0A(%0D%0A−%0D%0A1%0D%0A,%0D%0A5%0D%0A)%0D%0A, R(2,8)%0D%0A𝑅%0D%0A(%0D%0A2%0D%0A,%0D%0A8%0D%0A)%0D%0A, S(4,4)%0D%0A𝑆%0D%0A(%0D%0A4%0D%0A,%0D%0A4%0D%0A)%0D%0A, and T(1,1)%0D%0A𝑇%0D%0A(%0D%0A1%0D%0A,%0D%0A1%0D%0A)%0D%0A. The measure of Angle Q is 108°%0D%0A108%0D%0A°%0D%0A and the measure of angle R is 108−2x%0D%0A108%0D%0A−%0D%0A2%0D%0A𝑥%0D%0A. In 3-5 sentences, determine if quadrilateral QRST is a parallelogram, and if so, solve for x.%E2%80%8B(4 points)

Answer 1

Compute side vectors: QR = (3,3), ST = (−3,−3) and RS = (2,−4), QT = (2,−4), so opposite sides are parallel and equal; therefore QRST is a parallelogram. In a parallelogram adjacent angles are supplementary, so 108° + (108° − 2x) = 180°. Solving 216 − 2x = 180 gives 2x = 36 and x = 18.