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)
Quadrilateral QRST is NOT a parallelogram because there are NOT two pairs of parallel lines
Item
Quadrilateral QRST is a parallelogram because there are two pairs of parallel lines. The value x = 0 because the angles are congruent
Item 13
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 four sides. The value x = 108 because the angles are congruent
Determine if quadrilateral QRST is a parallelogram, and if so, solve for x.
(1 point)
Quadrilateral QRST is NOT a parallelogram because there are NOT two pairs of parallel lines
Item
Quadrilateral QRST is a parallelogram because there are two pairs of parallel lines. The value x = 0 because the angles are congruent
Item 13
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 four sides. The value x = 108 because the angles are congruent
Answers
GPT-4o mini
To determine whether quadrilateral QRST is a parallelogram and to solve for \( x \), we need to analyze the angles and the properties of the quadrilateral.
1. **Given Angles**:
- Measure of angle \( Q = 108^\circ \)
- Measure of angle \( R = (108 - 2x)^\circ \)
2. **Property of a Parallelogram**:
In a parallelogram, opposite angles are equal, and consecutive angles are supplementary (sum to \( 180^\circ \)).
3. **Finding the Condition for Angles**:
Since \( Q \) and \( R \) are consecutive angles, they must add up to \( 180^\circ \):
\[
\angle Q + \angle R = 180^\circ
\]
Substituting the values:
\[
108^\circ + (108 - 2x)^\circ = 180^\circ
\]
Simplifying:
\[
216 - 2x = 180
\]
\[
-2x = 180 - 216
\]
\[
-2x = -36
\]
\[
x = 18
\]
4. **Conclusion**:
The value of \( x \) is \( 18 \).
Now, let's summarize if quadrilateral QRST is a parallelogram:
- Since we found \( x \) such that the angles are supplementary, the angles \( Q \) and \( R \) meet the criteria for being in a parallelogram.
Thus, the correct statement is:
**Quadrilateral QRST is a parallelogram because there are two pairs of parallel lines. The value of \( x = 18 \) because the angles are supplementary.**
1. **Given Angles**:
- Measure of angle \( Q = 108^\circ \)
- Measure of angle \( R = (108 - 2x)^\circ \)
2. **Property of a Parallelogram**:
In a parallelogram, opposite angles are equal, and consecutive angles are supplementary (sum to \( 180^\circ \)).
3. **Finding the Condition for Angles**:
Since \( Q \) and \( R \) are consecutive angles, they must add up to \( 180^\circ \):
\[
\angle Q + \angle R = 180^\circ
\]
Substituting the values:
\[
108^\circ + (108 - 2x)^\circ = 180^\circ
\]
Simplifying:
\[
216 - 2x = 180
\]
\[
-2x = 180 - 216
\]
\[
-2x = -36
\]
\[
x = 18
\]
4. **Conclusion**:
The value of \( x \) is \( 18 \).
Now, let's summarize if quadrilateral QRST is a parallelogram:
- Since we found \( x \) such that the angles are supplementary, the angles \( Q \) and \( R \) meet the criteria for being in a parallelogram.
Thus, the correct statement is:
**Quadrilateral QRST is a parallelogram because there are two pairs of parallel lines. The value of \( x = 18 \) because the angles are supplementary.**