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.