Question

the angle measures of quadrilateral rstu are shown. m∠r=(2x)° m∠s=(3x - 35)° m∠t=(x + 35)°. the measure of angle u is unknown. can quadrilateral rstu be a parallelogram? yes, opposite angles r and t are congruent to each other if x = 35. yes, consecutive angles r and s are congruent to each other if x = 35. no, if x = 35, all three given angles measure 70°. the fourth angle would measure 150°. no, if x = 35, the three given angle measures make it impossible for the figure to be a quadrilateral.

Explanation:

Step1: Recall parallelogram angle - property

In a parallelogram, opposite angles are congruent and the sum of interior angles of a quadrilateral is 360°.

Step2: Set up an equation for opposite angles

If \(R\) and \(T\) are opposite angles in a parallelogram, then \(m\angle R=m\angle T\). So, \(2x=x + 35\). Solving for \(x\):
\[

$$\begin{align*} 2x-x&=35\\ x&=35 \end{align*}$$

\]

Step3: Find the measure of each given angle when \(x = 35\)

When \(x = 35\), \(m\angle R=2x=2\times35 = 70^{\circ}\), \(m\angle S=3x-35=3\times35 - 35=105 - 35=70^{\circ}\), \(m\angle T=x + 35=35+35 = 70^{\circ}\)

Step4: Find the measure of the fourth - angle

Let \(m\angle U=y\). Since the sum of the interior angles of a quadrilateral \(m\angle R+m\angle S+m\angle T+m\angle U = 360^{\circ}\), substituting the values of \(m\angle R\), \(m\angle S\), and \(m\angle T\): \(70+70 + 70+y=360\). Then \(y=360-(70 + 70+70)=150^{\circ}\)
In a parallelogram, consecutive angles are supplementary (\(m\angle R+m\angle S = 180^{\circ}\) for adjacent angles \(R\) and \(S\)), but here when \(x = 35\), \(m\angle R+m\angle S=70 + 70=140^{\circ}
eq180^{\circ}\)

Answer:

No, if \(x = 35\), all three given angles measure \(70^{\circ}\). The fourth angle would measure \(150^{\circ}\).