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.​(4 points)

Answer 1

To determine if quadrilateral QRST is a parallelogram, we can use the properties of the angles. The sum of the internal angles of a quadrilateral is 360°. Given that angle Q measures 108°, we can express angle R as \( 108 - 2x \). Let angle S and angle T be \( s \) and \( t \), respectively. Since we don't have their measures, we can set up the equation:

\[ 108 + (108 - 2x) + s + t = 360 \] \[ 216 - 2x + s + t = 360 \] \[ s + t = 144 + 2x \]

For QRST to be a parallelogram, opposite angles must be equal; hence angles Q and S and angles R and T must also be equal. If we suppose \( s = 108 \) (opposite to Q) and \( t = 108 - 2x \) (opposite to R), we can solve for \( x \):

If \( s = 108 \), then: \[ t = 108 - 2x \] Equating \( t \) with \( 108 \): \[ 108 - 2x = 108 \] \[ -2x = 0 \] \[ x = 0 \]

With \( x = 0 \), angle R also measures \( 108° \), making the opposite angles equal. Therefore, with equal opposite angles, quadrilateral QRST is indeed a parallelogram.