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: Recall properties of parallelograms

In a parallelogram, consecutive angles are supplementary (sum to \(180^\circ\)) and opposite angles are equal.
Given \(\angle S = 70^\circ\).

Step2: Find \(\angle R\)

\(\angle S\) and \(\angle R\) are consecutive angles, so \(m\angle S + m\angle R = 180^\circ\).
Substitute \(m\angle S = 70^\circ\): \(70^\circ + m\angle R = 180^\circ\)
Solve for \(m\angle R\): \(m\angle R = 180^\circ - 70^\circ = 110^\circ\)

Step3: Find \(\angle T\)

\(\angle S\) and \(\angle T\) are opposite angles? No, wait, \(\angle S\) and \(\angle U\) are opposite? Wait, in parallelogram \(RSTU\), vertices are \(R, S, T, U\) in order. So \(RS \parallel TU\) and \(RT \parallel SU\)? Wait, no, sides: \(RS\) and \(TU\) are parallel, \(ST\) and \(RU\) are parallel. So \(\angle S\) and \(\angle T\) are consecutive? Wait, no, \(\angle S\) and \(\angle R\) are consecutive (adjacent), \(\angle S\) and \(\angle T\) are... Wait, let's label the parallelogram: \(R\) connected to \(S\) and \(U\), \(S\) connected to \(R\) and \(T\), \(T\) connected to \(S\) and \(U\), \(U\) connected to \(T\) and \(R\). So sides: \(RS\), \(ST\), \(TU\), \(UR\). So \(\angle S\) is between \(RS\) and \(ST\), \(\angle R\) is between \(RS\) and \(UR\), \(\angle T\) is between \(ST\) and \(TU\), \(\angle U\) is between \(TU\) and \(UR\). So in a parallelogram, opposite angles are equal: \(\angle S = \angle U\), \(\angle R = \angle T\). Consecutive angles are supplementary: \(\angle S + \angle R = 180^\circ\), \(\angle R + \angle U = 180^\circ\), etc.
So \(\angle S = 70^\circ\), so \(\angle U = \angle S = 70^\circ\) (opposite angles). Then \(\angle R = 180^\circ - 70^\circ = 110^\circ\) (consecutive to \(\angle S\)), and \(\angle T = \angle R = 110^\circ\) (opposite angles). Wait, no, wait the options: let's check the options.
Wait the options:

1. \(m\angle R = 70^\circ\), \(m\angle T = 110^\circ\), \(m\angle U = 110^\circ\)
2. \(m\angle R = 110^\circ\), \(m\angle T = 110^\circ\), \(m\angle U = 70^\circ\)
3. \(m\angle R = 110^\circ\), \(m\angle T = 70^\circ\), \(m\angle U = 110^\circ\)
4. \(m\angle R = 70^\circ\), \(m\angle T = 110^\circ\), \(m\angle U = 70^\circ\)

Wait, let's re - examine. In parallelogram \(RSTU\), the order of vertices is \(R - S - T - U - R\). So \(RS \parallel TU\) and \(ST \parallel RU\). So \(\angle S\) and \(\angle T\) are consecutive angles (since \(S - T\) is a side, and \(S - R\) and \(T - U\) are parallel). Wait, no, consecutive angles are adjacent angles. So \(\angle S\) is adjacent to \(\angle R\) and \(\angle T\). \(\angle R\) is adjacent to \(\angle S\) and \(\angle U\). \(\angle T\) is adjacent to \(\angle S\) and \(\angle U\). \(\angle U\) is adjacent to \(\angle R\) and \(\angle T\).

So opposite angles: \(\angle S=\angle U\), \(\angle R = \angle T\). Consecutive angles: \(\angle S+\angle R = 180^\circ\), \(\angle R+\angle U=180^\circ\), \(\angle U+\angle T = 180^\circ\), \(\angle T+\angle S=180^\circ\).

Given \(\angle S = 70^\circ\), so \(\angle U=\angle S = 70^\circ\) (opposite angles). Then \(\angle R=180^\circ - \angle S=180 - 70 = 110^\circ\) (consecutive to \(\angle S\)). And \(\angle T=\angle R = 110^\circ\) (opposite angles). So \(m\angle R = 110^\circ\), \(m\angle T = 110^\circ\), \(m\angle U = 70^\circ\), which is the second option.

Answer:

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