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)°.
Determine if quadrilateral QRST is a parallelogram, and if so, solve for x.
(1 point)
Responses
Quadrilateral QRST is a parallelogram because there are four sides. The value x = 108 because the angles are congruent.
Quadrilateral QRST is a parallelogram because there are four sides. The value x = 108 because the angles are congruent.
Quadrilateral QRST is a parallelogram because there are two pairs of parallel lines. The value x = 0 because the angles are congruent.
Quadrilateral QRST is a parallelogram because there are two pairs of parallel lines. The value x = 0 because the angles are congruent.
Quadrilateral QRST is a parallelogram because there are two pairs of parallel lines. The value x = 18 because the angles are supplementary.
Quadrilateral QRST is a parallelogram because there are two pairs of parallel lines. The value x = 18 because the angles are supplementary.
Quadrilateral QRST is NOT a parallelogram because there are NOT two pairs of parallel lines.
Determine if quadrilateral QRST is a parallelogram, and if so, solve for x.
(1 point)
Responses
Quadrilateral QRST is a parallelogram because there are four sides. The value x = 108 because the angles are congruent.
Quadrilateral QRST is a parallelogram because there are four sides. The value x = 108 because the angles are congruent.
Quadrilateral QRST is a parallelogram because there are two pairs of parallel lines. The value x = 0 because the angles are congruent.
Quadrilateral QRST is a parallelogram because there are two pairs of parallel lines. The value x = 0 because the angles are congruent.
Quadrilateral QRST is a parallelogram because there are two pairs of parallel lines. The value x = 18 because the angles are supplementary.
Quadrilateral QRST is a parallelogram because there are two pairs of parallel lines. The value x = 18 because the angles are supplementary.
Quadrilateral QRST is NOT a parallelogram because there are NOT two pairs of parallel lines.
Answers
GPT-4o mini
To determine whether quadrilateral QRST is a parallelogram and to find the value of \( x \), we will first consider the properties of angles in a quadrilateral.
In any quadrilateral, the sum of the interior angles is \( 360^\circ \). We know that:
- \( \angle Q = 108^\circ \)
- \( \angle R = (108 - 2x)^\circ \)
Let’s denote the measures of angles \( S \) and \( T \) as \( \angle S \) and \( \angle T \) respectively. We will now apply the angle sum property of quadrilaterals:
\[
\angle Q + \angle R + \angle S + \angle T = 360^\circ
\]
Substituting the known angles gives us:
\[
108 + (108 - 2x) + \angle S + \angle T = 360
\]
Simplifying further:
\[
216 - 2x + \angle S + \angle T = 360
\]
Rearranging the equation provides:
\[
\angle S + \angle T = 360 - 216 + 2x
\]
\[
\angle S + \angle T = 144 + 2x
\]
Now, for QRST to be a parallelogram, the angles opposite each other must be equal, so we consider the angles Q and S, and R and T.
Let's say \( \angle S \) is equal to \( \angle Q \) and \( \angle T \) is equal to \( \angle R \):
If \( \angle S = \angle Q = 108^\circ \) (so its opposite angle is equal), then
\[
\angle T = (108 - 2x).
\]
Since \( \angle T \) also equals \( R \):
\[
\angle T = 108 - 2x.
\]
Setting these equal gives us:
\[
\angle T + \angle R = 180^\circ \quad \text{(since they should be supplementary)}
\]
Thus,
\[
108 + (108 - 2x) = 180
\]
\[
216 - 2x = 180
\]
\[
-2x = 180 - 216
\]
\[
-2x = -36
\]
\[
x = 18.
\]
Thus, we have found \( x = 18 \).
Now we assess our options considering Q and S as opposite angles:
Since QRST has opposite angles equal, we can conclude that quadrilateral QRST is indeed a parallelogram.
The correct response is:
**"Quadrilateral QRST is a parallelogram because there are two pairs of parallel lines. The value \( x = 18 \) because the angles are supplementary."**
In any quadrilateral, the sum of the interior angles is \( 360^\circ \). We know that:
- \( \angle Q = 108^\circ \)
- \( \angle R = (108 - 2x)^\circ \)
Let’s denote the measures of angles \( S \) and \( T \) as \( \angle S \) and \( \angle T \) respectively. We will now apply the angle sum property of quadrilaterals:
\[
\angle Q + \angle R + \angle S + \angle T = 360^\circ
\]
Substituting the known angles gives us:
\[
108 + (108 - 2x) + \angle S + \angle T = 360
\]
Simplifying further:
\[
216 - 2x + \angle S + \angle T = 360
\]
Rearranging the equation provides:
\[
\angle S + \angle T = 360 - 216 + 2x
\]
\[
\angle S + \angle T = 144 + 2x
\]
Now, for QRST to be a parallelogram, the angles opposite each other must be equal, so we consider the angles Q and S, and R and T.
Let's say \( \angle S \) is equal to \( \angle Q \) and \( \angle T \) is equal to \( \angle R \):
If \( \angle S = \angle Q = 108^\circ \) (so its opposite angle is equal), then
\[
\angle T = (108 - 2x).
\]
Since \( \angle T \) also equals \( R \):
\[
\angle T = 108 - 2x.
\]
Setting these equal gives us:
\[
\angle T + \angle R = 180^\circ \quad \text{(since they should be supplementary)}
\]
Thus,
\[
108 + (108 - 2x) = 180
\]
\[
216 - 2x = 180
\]
\[
-2x = 180 - 216
\]
\[
-2x = -36
\]
\[
x = 18.
\]
Thus, we have found \( x = 18 \).
Now we assess our options considering Q and S as opposite angles:
Since QRST has opposite angles equal, we can conclude that quadrilateral QRST is indeed a parallelogram.
The correct response is:
**"Quadrilateral QRST is a parallelogram because there are two pairs of parallel lines. The value \( x = 18 \) because the angles are supplementary."**