Question

look at this diagram;
if (overleftrightarrow{qs}) and (overleftrightarrow{tv}) are parallel lines and (mangle sru = 136^circ), what is (mangle t uw)?
(square^circ)

Explanation:

Step1: Identify supplementary angles

Since \(\angle SRU\) and its adjacent angle on the straight line \(QS\) (let's say \(\angle QRU\)) are supplementary, but actually, \(\angle SRU\) and \(\angle TRU\) (or the corresponding angle with \(\angle TUW\)): Wait, more accurately, \(QS \parallel TV\), and the transversal is \(PW\) (the line with points \(P, R, U, W\)). So \(\angle SRU\) and \(\angle TUW\) are same - side interior angles? No, wait, \(\angle SRU\) and the angle adjacent to \(\angle TUW\) on the straight line \(TV\) (at point \(U\)): Actually, \(\angle SRU\) and \(\angle TUW\) are supplementary? Wait, no. Let's think again. The sum of a linear pair is \(180^{\circ}\), and since \(QS\parallel TV\), alternate - interior or same - side? Wait, \(\angle SRU\) and \(\angle TUW\): Let's see, \(\angle SRU = 136^{\circ}\), and \(\angle TUW\) and \(\angle SRU\) are supplementary? Wait, no, actually, \(\angle SRU\) and the angle \(\angle QRU\) (which is adjacent to \(\angle SRU\) on line \(QS\)) is \(180 - 136=44^{\circ}\), and since \(QS\parallel TV\), by alternate - interior angles, \(\angle TUW=\angle QRU\). Wait, let's do it step by step.

Step1: Find the supplementary angle of \(\angle SRU\)

A linear pair of angles sums to \(180^{\circ}\). So if \(m\angle SRU = 136^{\circ}\), then the angle adjacent to it on the straight line \(QS\) (let's call it \(\angle QRU\)) is \(180^{\circ}-m\angle SRU\).
\[m\angle QRU=180^{\circ}- 136^{\circ}=44^{\circ}\]

Step2: Use the property of parallel lines

Since \(QS\parallel TV\) and the transversal is the line containing \(R, U\) (the line with \(P, R, U, W\)), \(\angle TUW\) and \(\angle QRU\) are alternate - interior angles. Alternate - interior angles are equal when two parallel lines are cut by a transversal. So \(m\angle TUW=m\angle QRU = 44^{\circ}\)

Answer:

\(44\)