Question

look at this diagram:
if \\( \overleftrightarrow { o q } \\) and \\( \overleftrightarrow { r t } \\) are parallel lines and \\( m \angle o p n = 113 ^ { \circ } \\), what is \\( m \angle r s u \\)?

Explanation:

Step1: Find the measure of ∠SPT

Since ∠OPN and ∠SPT are vertical angles, and vertical angles are equal.
So, \(m\angle SPT=m\angle OPN = 113^{\circ}\)

Step2: Use the property of parallel lines

Since \(\overleftrightarrow{OQ}\) and \(\overleftrightarrow{RT}\) are parallel lines and \(UN\) is a transversal.
\(\angle RSU\) and \(\angle SPT\) are supplementary angles (consecutive interior angles).
The sum of consecutive - interior angles is \(180^{\circ}\). Let \(m\angle RSU=x\).
We have the equation \(x + 113^{\circ}=180^{\circ}\)

Step3: Solve for \(x\)

Subtract \(113^{\circ}\) from both sides of the equation:
\(x=180^{\circ}- 113^{\circ}\)
\(x = 67^{\circ}\)

Answer:

\(67\)