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

Is this a parallelogram?

Answer 1

Yes.

Check slopes of opposite sides:
- slope QR = (8−5)/(2−(−1)) = 3/3 = 1 and slope ST = (1−4)/(1−4) = −3/−3 = 1 → QR ∥ ST.
- slope RS = (4−8)/(4−2) = −4/2 = −2 and slope TQ = (5−1)/(−1−1) = 4/−2 = −2 → RS ∥ TQ.

Since both pairs of opposite sides are parallel, QRST is a parallelogram.

(If you use the angle info: adjacent angles in a parallelogram are supplementary, so 108 + (108 − 2x) = 180 → x = 18.)