Question

look at this diagram:
if \\( \overleftrightarrow { m o } \\) and \\( \overleftrightarrow { p r } \\) are parallel lines and \\( m \angle r q n = 66 ^ { \circ } \\), what is \\( m \angle m n q? \\)

Explanation:

Step1: Use the property of consecutive interior angles

When two parallel lines are cut by a transversal, consecutive interior angles are supplementary. Here, \(\overleftrightarrow{MO}\) and \(\overleftrightarrow{PR}\) are parallel lines and \(LS\) is the transversal. \(\angle RQN\) and \(\angle MNQ\) are consecutive interior angles.

Step2: Apply the supplementary - angle formula

If two angles \(\alpha\) and \(\beta\) are supplementary, then \(\alpha+\beta = 180^{\circ}\). Let \(\alpha=\angle RQN = 66^{\circ}\) and \(\beta=\angle MNQ\). So, \(m\angle MNQ=180^{\circ}-m\angle RQN\).
Substitute \(m\angle RQN = 66^{\circ}\) into the formula: \(m\angle MNQ=180 - 66\).
\(m\angle MNQ = 114^{\circ}\)

Answer:

\(114\)