Question

look at this diagram: if $overleftrightarrow{np}$ and $overleftrightarrow{qs}$ are parallel lines and $mangle qro = 46^circ$, what is $mangle nor$?

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Explanation:

Step1: Identify the relationship

Since \( \overleftrightarrow{NP} \parallel \overleftrightarrow{QS} \) and \( \overleftrightarrow{TM} \) is a transversal, \( \angle QRO \) and \( \angle NOR \) are same - side interior angles? Wait, no. Wait, actually, \( \angle QRO \) and \( \angle NOR \) are consecutive interior angles? Wait, no, let's look at the lines. \( QS \) and \( NP \) are parallel, and \( TM \) is a transversal. \( \angle QRO \) and \( \angle NOR \): Wait, \( \angle QRO \) and \( \angle NOR \) are supplementary? Wait, no, wait, actually, if we consider the parallel lines \( QS \) and \( NP \), and transversal \( TM \), then \( \angle QRO \) and \( \angle NOR \) are same - side interior angles? Wait, no, let's think again. Wait, \( \angle QRO \) and \( \angle NOR \): Let's see the positions. \( QS \) is parallel to \( NP \), and the transversal is \( TM \). So \( \angle QRO \) and \( \angle NOR \) are same - side interior angles? Wait, no, actually, \( \angle QRO \) and \( \angle NOR \) are supplementary? Wait, no, wait, maybe they are same - side interior angles, but wait, no, let's check the angles. Wait, if two parallel lines are cut by a transversal, same - side interior angles are supplementary. But wait, \( \angle QRO = 46^{\circ}\), then \( \angle NOR=180 - 46=134^{\circ}\)? Wait, no, wait, maybe I made a mistake. Wait, no, let's look at the diagram again. Wait, \( QS \) and \( NP \) are vertical lines (parallel), and \( TM \) is a slant line. So \( \angle QRO \) and \( \angle NOR \): Let's see, \( \angle QRO \) is at \( R \) between \( QS \) and \( TM \), and \( \angle NOR \) is at \( O \) between \( NP \) and \( TM \). Since \( QS \parallel NP \), and \( TM \) is a transversal, \( \angle QRO \) and \( \angle NOR \) are same - side interior angles, so they are supplementary. So \( m\angle NOR = 180^{\circ}-m\angle QRO\).

Step2: Calculate \( m\angle NOR \)

Given \( m\angle QRO = 46^{\circ} \), then \( m\angle NOR=180^{\circ}- 46^{\circ}=134^{\circ} \). Wait, no, wait, maybe they are alternate interior angles? Wait, no, alternate interior angles are equal. Wait, maybe I misidentified the angles. Wait, let's re - examine the diagram. \( QS \) and \( NP \) are parallel. The transversal is \( TM \). \( \angle QRO \) and \( \angle NOR \): Let's see the vertices. \( R \) is on \( QS \), \( O \) is on \( NP \), and \( T - R - O - M \) is the transversal. So \( \angle QRO \) and \( \angle NOR \): are they same - side interior angles? Wait, same - side interior angles are on the same side of the transversal and inside the two parallel lines. \( QS \) and \( NP \) are the two parallel lines, \( TM \) is the transversal. \( \angle QRO \) is inside \( QS \) and \( TM \), \( \angle NOR \) is inside \( NP \) and \( TM \), and they are on the same side of \( TM \). So they are same - side interior angles, so they are supplementary. So \( m\angle NOR = 180 - 46=134 \). Wait, but maybe I got it wrong. Wait, no, maybe they are consecutive interior angles, which are supplementary. So the calculation is \( 180 - 46 = 134 \).

Answer:

\( 134 \)