Question

look at this diagram:
diagram with lines oq, rt (parallel), and transversal nu intersecting oq at p and rt at s; points n, p, s, u on transversal; q, p, o on oq; t, s, r on rt
if \overleftrightarrow{oq} and \overleftrightarrow{rt} are parallel lines and m\angle tsu = 129\degree, what is m\angle opn?

Explanation:

Step1: Identify Supplementary Angles

Since \( \angle TSU \) and its adjacent angle on line \( RT \) are supplementary (they form a linear pair), we first find the adjacent angle. Let \( \angle TSR \) be the adjacent angle to \( \angle TSU \). Then \( m\angle TSR = 180^\circ - m\angle TSU \). Given \( m\angle TSU = 129^\circ \), so \( m\angle TSR = 180^\circ - 129^\circ = 51^\circ \).

Step2: Use Corresponding Angles (Parallel Lines)

Since \( OQ \parallel RT \) and \( NU \) is a transversal, \( \angle OPN \) and \( \angle TSR \) are corresponding angles. Corresponding angles are equal when lines are parallel. So \( m\angle OPN = m\angle TSR = 51^\circ \).

Answer:

\( 51^\circ \)