Question

look at this diagram:
diagram of two parallel lines mo and pr cut by a transversal ls, with points m, n, o on mo; p, q, r on pr; and l, n, s, q on ls
if \\( \overleftrightarrow{mo} \\) and \\( \overleftrightarrow{pr} \\) are parallel lines and \\( m\angle mnq = 43^\circ \\), what is \\( m\angle rqn \\)?
\\( \square^\circ \\)
submit

Explanation:

Step1: Identify angle relationship

Since \( \overleftrightarrow{MO} \parallel \overleftrightarrow{PR} \) and \( \overleftrightarrow{LS} \) is a transversal, \( \angle MNQ \) and \( \angle RQN \) are same - side interior angles? Wait, no, wait. Wait, actually, \( \angle MNQ \) and \( \angle RQN \): let's check the positions. \( MO \parallel PR \), transversal \( LNQS \). \( \angle MNQ \) and \( \angle RQN \): are they supplementary? Wait, no, wait. Wait, \( \angle MNQ \) and \( \angle RQN \): if we look at the lines, \( MO \parallel PR \), and the transversal is \( NQ \) (part of \( LS \)). Wait, actually, \( \angle MNQ \) and \( \angle RQN \) are same - side interior angles? Wait, no, let's think again. Wait, \( MO \) is parallel to \( PR \), and the transversal is \( LNQS \). So \( \angle MNQ \) and \( \angle RQN \): are they consecutive interior angles? Wait, no, \( \angle MNQ \) is at \( N \) between \( MO \) and \( LN \), and \( \angle RQN \) is at \( Q \) between \( PR \) and \( LN \). Wait, actually, \( \angle MNQ \) and \( \angle RQN \) are same - side interior angles? Wait, no, maybe they are supplementary? Wait, no, wait. Wait, if \( MO \parallel PR \), and the transversal is \( LN \), then \( \angle MNQ \) and \( \angle RQN \): let's see, \( \angle MNQ \) and \( \angle RQN \) are same - side interior angles? Wait, no, \( \angle MNQ \) is on \( MO \), and \( \angle RQN \) is on \( PR \), with the transversal \( LN \). Wait, actually, \( \angle MNQ \) and \( \angle RQN \) are same - side interior angles, so they should be supplementary? Wait, no, that can't be. Wait, no, wait, maybe I made a mistake. Wait, \( \angle MNQ \) and \( \angle RQN \): let's check the sum. Wait, if \( MO \parallel PR \), then consecutive interior angles are supplementary. But \( \angle MNQ \) and \( \angle RQN \): are they consecutive interior angles? Let's see, the two parallel lines are \( MO \) and \( PR \), the transversal is \( LN \). So \( \angle MNQ \) is between \( MO \) and \( LN \), and \( \angle RQN \) is between \( PR \) and \( LN \), on the same side of the transversal. So they are consecutive interior angles, so they should be supplementary? Wait, but that would mean \( m\angle RQN=180 - 43 = 137 \)? Wait, no, that can't be right. Wait, maybe I got the angle wrong. Wait, no, wait, maybe \( \angle MNQ \) and \( \angle RQN \) are same - side interior angles? Wait, no, maybe they are alternate interior angles? Wait, no, alternate interior angles would be \( \angle MNQ \) and \( \angle PQN \). Wait, \( MO \parallel PR \), transversal \( LN \), so \( \angle MNQ \) and \( \angle PQN \) are alternate interior angles, so they are equal. But \( \angle RQN \) and \( \angle PQN \) are supplementary, because they form a linear pair. Wait, \( \angle PQN + \angle RQN=180^{\circ} \), since they are adjacent angles on a straight line. Wait, but \( \angle MNQ=\angle PQN = 43^{\circ} \) (alternate interior angles), so \( \angle RQN = 180 - 43=137^{\circ} \)? Wait, no, that seems off. Wait, maybe I misidentified the angles. Wait, let's look at the diagram again. \( MO \) is a horizontal line (left - right), \( PR \) is also horizontal (left - right), parallel. The transversal is a line going from bottom left (S) to top right (L), passing through N (on MO) and Q (on PR). \( \angle MNQ \) is the angle at N, between M (left on MO) and NQ (the transversal). \( \angle RQN \) is the angle at Q, between R (right on PR) and NQ (the transversal). So \( MO \parallel PR \), transversal NQ. So \( \angle MNQ \) and \( \angle RQN \): are they same - side interior angles? Let's see, \( MO \) and \( PR \) are parallel, transversal NQ. The interior angles on the same side of the transversal: \( \angle MNQ \) is on MO, above the transversal? Wait, no, \( MO \) is horizontal, NQ is going up to the right. So \( \angle MNQ \) is the angle between MO (left - right) and NQ (up - right), so it's a small angle (43 degrees). \( \angle RQN \) is the angle between PR (left - right) and NQ (up - right), on the right side of NQ. Wait, no, PR is a horizontal line, Q is on PR, R is to the right of Q, NQ is going up to the left? Wait, no, the transversal is going from S (bottom left) to L (top right), so NQ is going from N (on MO) to Q (on PR), so the direction is up - right. So at point Q, PR is horizontal (left - right), NQ is going up - right, so \( \angle RQN \) is the angle between PR (right - left? No, PR is from P (left) to R (right), so at Q, PR is going from Q to R (right), and NQ is going from Q to N (up - left). Wait, oh! I see my mistake. NQ is going from N (on MO) to Q (on PR), so the direction from Q to N is up - left, and from Q to R is right (along PR). So \( \angle RQN \) is the angle between RQ (right along PR) and QN (up - left). So \( MO \parallel PR \), transversal QN. So \( \angle MNQ \) (at N, between MQ? No, M is left on MO, N is on MO, so MN is left - right, NQ is up - right. \( \angle MNQ \) is between MN (left) and NQ (up - right). \( \angle RQN \) is between RQ (right) and QN (up - left). So these two angles: \( \angle MNQ \) and \( \angle RQN \): are they same - side interior angles? Wait, no, actually, \( \angle MNQ \) and \( \angle RQN \) are same - side interior angles? Wait, no, let's use the consecutive interior angles theorem. Consecutive interior angles are supplementary. So \( \angle MNQ \) and \( \angle RQN \) are consecutive interior angles? Let's check: \( MO \parallel PR \), transversal QN. The interior angles on the same side of the transversal: \( \angle MNQ \) is an interior angle (between MO and PR, on the left side of the transversal), and \( \angle RQN \) is an interior angle (between MO and PR, on the left side of the transversal? No, that doesn't make sense. Wait, maybe \( \angle MNQ \) and \( \angle RQN \) are same - side interior angles, so they should be supplementary. So \( m\angle RQN = 180 - m\angle MNQ=180 - 43 = 137^{\circ} \). Wait, but that seems like a large angle. Alternatively, maybe I got the angle type wrong. Wait, maybe \( \angle MNQ \) and \( \angle RQN \) are same - side interior angles, so they are supplementary. So the calculation is \( 180 - 43 = 137 \).

Step2: Calculate the measure of \( \angle RQN \)

Since \( \overleftrightarrow{MO} \parallel \overleftrightarrow{PR} \) and \( \overleftrightarrow{NQ} \) is a transversal, \( \angle MNQ \) and \( \angle RQN \) are same - side interior angles. Same - side interior angles are supplementary, meaning their measures add up to \( 180^{\circ} \).

We know that \( m\angle MNQ = 43^{\circ} \). Let \( m\angle RQN=x \). Then:

\( m\angle MNQ + m\angle RQN=180^{\circ} \)

Substitute \( m\angle MNQ = 43^{\circ} \) into the equation:

\( 43^{\circ}+x = 180^{\circ} \)

Solve for \( x \):

\( x=180^{\circ}- 43^{\circ}=137^{\circ} \)

Answer:

\( 137 \)