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 degrees 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 can use the property that in a parallelogram, the sum of opposite angles must be equal to 180 degrees. We are given that the measure of angle Q is 108 degrees and the measure of angle R is \(108 - 2x\).

According to the properties of a parallelogram, we have:

\[
\angle Q + \angle R = 180^\circ
\]

Substituting the known angles:

\[
108 + (108 - 2x) = 180
\]

Simplifying this equation:

\[
216 - 2x = 180
\]
\[
-2x = 180 - 216
\]
\[
-2x = -36
\]
\[
x = 18
\]

Since we found a consistent solution for \(x\), we conclude that quadrilateral QRST is indeed a parallelogram, and \(x = 18\).