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.​

Answer 1

To determine if quadrilateral QRST is a parallelogram, we need to check if the consecutive angles sum to 180°. We know that angle Q measures 108°, and angle R measures \( 108 - 2x \). Therefore, the sum of angles Q and R is \( 108 + (108 - 2x) = 216 - 2x \). For QRST to be a parallelogram, we must have \( 216 - 2x = 180 \).

Solving for \( x \):
\[
216 - 2x = 180 \\
2x = 216 - 180 \\
2x = 36 \\
x = 18
\]

Since we found a valid value for \( x \), quadrilateral QRST can be a parallelogram if angles Q and R sum appropriately. Thus, \( x = 18 \) indicates it is a parallelogram, assuming the properties of opposite angles and sides are satisfied.