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)°

Question

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

Answer 1

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.