Question

what are the missing angle measures in parallelogram rstu?
○ ( mangle r = 70^{circ}, mangle t = 110^{circ}, mangle u = 110^{circ} )
○ ( mangle r = 110^{circ}, mangle t = 110^{circ}, mangle u = 70^{circ} )
○ ( mangle r = 110^{circ}, mangle t = 70^{circ}, mangle u = 110^{circ} )
○ ( mangle r = 70^{circ}, mangle t = 110^{circ}, mangle u = 70^{circ} )

Explanation:

Step1: Properties of parallelogram (opposite angles are equal)

In parallelogram \( RSTU\), \(\angle S\) and \(\angle U\) are not opposite. \(\angle S\) and \(\angle T\) are consecutive. \(\angle R\) and \(\angle T\) are consecutive. \(\angle R\) and \(\angle S\) are consecutive. Since \( \angle S=70^{\circ}\), and in a parallelogram opposite angles are equal. So \( \angle R\) and \( \angle T\) are consecutive angles. Consecutive angles in a parallelogram are supplementary (\(m\angle R + m\angle S=180^{\circ}\)).

$$m\angle R=180^{\circ}- 70^{\circ}=110^{\circ}$$

Step2: Opposite angles

Since \( \angle S = 70^{\circ}\), and in a parallelogram opposite angles are equal. So \(m\angle U=m\angle S = 70^{\circ}\)

Step3: Another pair of opposite angles

\(m\angle R=m\angle T\) (opposite angles of a parallelogram). But wait, no! Wait, consecutive angles: \(m\angle S + m\angle T=180^{\circ}\) (because \(RS\parallel TU\) and \(ST\) is a transversal). So \(m\angle T = 180^{\circ}-70^{\circ}=110^{\circ}\). Wait no, correction: Opposite angles: \(m\angle R=m\angle T\) (wrong, no. Wait, in parallelogram \(RSTU\), \( \angle R\) and \( \angle T\) are not opposite. \( \angle R\) and \( \angle U\) are not opposite. \( \angle S\) and \( \angle U\) are opposite (\(m\angle S = m\angle U = 70^{\circ}\)), \( \angle R\) and \( \angle T\) are opposite (\(m\angle R=m\angle T = 110^{\circ}\))

Wait, no, another approach:
In parallelogram \(RSTU\), \( \angle S\) and \( \angle U\) are opposite (\(m\angle S=m\angle U = 70^{\circ}\)), \( \angle R\) and \( \angle T\) are opposite. Also, \( \angle S\) and \( \angle R\) are consecutive (\(m\angle S + m\angle R=180^{\circ}\)), so \(m\angle R=110^{\circ}\), then \(m\angle T=m\angle R = 110^{\circ}\)

Answer:

\(m\angle R = 110^{\circ},m\angle T = 110^{\circ},m\angle U = 70^{\circ}\) (the second option)