Question

Quadrilateral QRST has vertices at Q(−1,5)
𝑄
(
−
1
,
5
)
, R(2,8)
𝑅
(
2
,
8
)
, S(4,4)
𝑆
(
4
,
4
)
, and T(1,1)
𝑇
(
1
,
1
)
. The measure of Angle Q is 108°
108
°
and the measure of angle R is 108−2x
108
−
2
𝑥
. In 3-5 sentences, determine if quadrilateral QRST is a parallelogram, and if so, solve for x.​(4 points)
simple and basic

Answer 1

To determine if quadrilateral QRST is a parallelogram, we need to check if the sum of the interior angles equals 360°. Given that angle Q measures 108° and angle R measures \(108° - 2x\), the measures of angles S and T are missing. However, for QRST to be a parallelogram, opposite angles must be equal.

Let's denote angle S as \(y\) and angle T as \(z\). If we assume angle S = angle R and angle T = angle Q (since opposite angles in a parallelogram are equal), we have:

- Angle Q = 108°
- Angle R = 108° - 2x
- Angle S = 108° - 2x
- Angle T = 108°

To find x, we can set up the equation:
\[ 108 + (108 - 2x) + (108 - 2x) + 108 = 360 \]

Simplifying this gives:
\[ 432 - 4x = 360 \]
\[ 4x = 72 \]
\[ x = 18 \]

Thus, if QRST is indeed a parallelogram, \(x\) equals 18.