Question

look at this diagram: if $\overleftrightarrow{km}$ and $\overleftrightarrow{np}$ are parallel lines and $m\angle noq = 48^\circ$, what is $m\angle klt$? $\square^\circ$

Explanation:

Step1: Identify Angle Relationship

Since \( KM \parallel NP \) and \( QJ \) is a transversal, \( \angle NOQ \) and \( \angle KLJ \) are same - side interior angles? Wait, no, actually, \( \angle NOQ \) and \( \angle KLJ \): Wait, \( \angle NOQ \) and \( \angle KLO \) would be alternate interior angles, but \( \angle KLJ \) and \( \angle KLO \) are supplementary? Wait, no, let's re - examine.

Wait, \( NP \parallel KM \), and the transversal is \( QJ \). \( \angle NOQ \) and \( \angle KLJ \): Wait, \( \angle NOQ \) and \( \angle KLJ \) are same - side interior angles? No, actually, \( \angle NOQ \) and \( \angle KLJ \): Wait, \( \angle NOQ \) and \( \angle KLO \) are alternate interior angles (since \( NP\parallel KM \), and \( QJ \) is transversal). But \( \angle KLO \) and \( \angle KLJ \) are supplementary? Wait, no, \( \angle KLO \) and \( \angle KLJ \): Wait, \( \angle NOQ = 48^{\circ}\), and since \( NP\parallel KM \), \( \angle NOQ \) and \( \angle KLJ \) are same - side interior angles? Wait, no, let's think again.

Wait, \( NP \) and \( KM \) are parallel, \( QJ \) is a transversal. \( \angle NOQ \) and \( \angle KLJ \): \( \angle NOQ \) is at \( O \) between \( NP \) and \( QJ \), \( \angle KLJ \) is at \( L \) between \( KM \) and \( QJ \). Since \( NP\parallel KM \), \( \angle NOQ \) and \( \angle KLJ \) are same - side interior angles? Wait, no, same - side interior angles are supplementary. Wait, no, maybe \( \angle NOQ \) and \( \angle KLJ \) are supplementary? Wait, no, let's check the positions.

Wait, \( \angle NOQ \) and \( \angle KLO \) are alternate interior angles, so \( \angle NOQ=\angle KLO = 48^{\circ}\). Then \( \angle KLO \) and \( \angle KLJ \) are supplementary (since they form a linear pair), so \( m\angle KLJ=180^{\circ}- 48^{\circ}=132^{\circ}\)? Wait, no, that can't be. Wait, maybe I got the angle relationship wrong.

Wait, no, let's look at the diagram again. \( NP \) and \( KM \) are parallel, vertical lines. \( QJ \) is a transversal. \( \angle NOQ \) is an angle between \( NP \) (vertical line) and \( QJ \) (the transversal). \( \angle KLJ \) is an angle between \( KM \) (vertical line) and \( QJ \) (transversal). Since \( NP\parallel KM \), \( \angle NOQ \) and \( \angle KLJ \) are same - side interior angles? Wait, no, same - side interior angles are on the same side of the transversal. Wait, \( \angle NOQ \) and \( \angle KLJ \): if \( NP \) and \( KM \) are parallel, and \( QJ \) is the transversal, then \( \angle NOQ \) and \( \angle KLJ \) are same - side interior angles, so they should be supplementary. Wait, but maybe \( \angle NOQ \) and \( \angle KLJ \) are corresponding angles? Wait, no, corresponding angles would be in the same position relative to the parallel lines and transversal.

Wait, let's start over. \( NP \parallel KM \), transversal \( QJ \). \( \angle NOQ \) and \( \angle KLJ \): \( \angle NOQ \) is at \( O \), between \( NP \) (up - down line) and \( QJ \) (slanted line). \( \angle KLJ \) is at \( L \), between \( KM \) (up - down line) and \( QJ \) (slanted line). Since \( NP\parallel KM \), these two angles are same - side interior angles. Same - side interior angles are supplementary, so \( m\angle KLJ = 180^{\circ}-m\angle NOQ\).

Step2: Calculate the Measure of \( \angle KLJ \)

We know that \( m\angle NOQ = 48^{\circ}\). So \( m\angle KLJ=180 - 48=132^{\circ}\)? Wait, no, that seems wrong. Wait, maybe \( \angle NOQ \) and \( \angle KLJ \) are alternate interior angles? Wait, no, alternate interior angles are equal. Wait, maybe I mixed up the angle names.

Wait, \( \angle NOQ \) and \( \angle KLO \) are alternate interior angles, so \( \angle NOQ=\angle KLO = 48^{\circ}\). Then \( \angle KLO \) and \( \angle KLJ \) are supplementary (linear pair), so \( m\angle KLJ = 180 - 48=132^{\circ}\). Yes, that makes sense. Because \( \angle KLO \) and \( \angle KLJ \) form a straight line, so they are supplementary. And since \( NP\parallel KM \), \( \angle NOQ \) (alternate interior to \( \angle KLO \)) is equal to \( \angle KLO \). Then \( \angle KLJ = 180^{\circ}-\angle KLO=180 - 48 = 132^{\circ}\).

Answer:

\(132\)