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 a 3-5 sentence paragraph, determine if quadrilateral QRST is a parallelogram, and if so, solve for x.​

Answer 1

To determine if quadrilateral QRST is a parallelogram, we can use the property that the sum of the interior angles of a quadrilateral is 360°. Given that angle Q measures 108° and angle R measures \(108 - 2x\), we can denote the measures of angles S and T as \(s\) and \(t\), respectively. Thus, we have the equation:

\[
108 + (108 - 2x) + s + t = 360
\]

Simplifying this gives:

\[
216 - 2x + s + t = 360 \quad \Rightarrow \quad s + t = 144 + 2x
\]

For QRST to be a parallelogram, opposite angles must be equal. Therefore, angle S must equal angle R, and angle T must equal angle Q, leading us to the equations:

\[
s = 108 - 2x \quad \text{and} \quad t = 108
\]

Substituting \(t = 108\) into the previous equation, we get:

\[
s + 108 = 144 + 2x \quad \Rightarrow \quad s = 144 + 2x - 108 \quad \Rightarrow \quad s = 36 + 2x
\]

Setting the equations for \(s\) equal, we have:

\[
36 + 2x = 108 - 2x
\]
\[
4x = 72 \quad \Rightarrow \quad x = 18
\]

Thus, QRST is a parallelogram, and \(x\) solves to 18.