look at this diagram: diagram of lines km, np parallel, transversal jq with points k, l, m on km; n, o, p on np; j, l, o, q on transversal. if $ \overleftrightarrow{km} $ and $ \overleftrightarrow{np} $ are parallel lines and $ m\angle klj = 130^\circ $, what is $ m\angle nol $?

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# Explanation:
## Step1: Identify the relationship between angles
Since $ KM \parallel NP $ and $ JQ $ is a transversal, $ \angle KLJ $ and $ \angle NOL $ are same - side interior angles? Wait, no. Wait, $ \angle KLJ $ and $ \angle MLO $ are vertical angles? No, actually, $ \angle KLJ $ and $ \angle NOL $: Let's see, $ KM \parallel NP $, and $ JQ $ is a transversal. $ \angle KLJ $ and $ \angle NOL $ are actually alternate - interior angles? Wait, no, $ \angle KLJ $ and $ \angle MLO $ are vertical angles, but $ \angle MLO $ and $ \angle NOL $: Wait, $ KM \parallel NP $, so consecutive interior angles are supplementary. Wait, $ \angle KLJ $ and $ \angle NOL $: Let's check the positions. $ \angle KLJ $ is at $ L $ between $ KL $ and $ LJ $, $ \angle NOL $ is at $ O $ between $ NO $ and $ OL $. Since $ KM \parallel NP $, and $ JQ $ is a transversal, $ \angle KLJ $ and $ \angle NOL $ are same - side interior angles? No, wait, $ \angle KLJ $ and $ \angle MLO $ are vertical angles, so $ m\angle MLO=m\angle KLJ = 130^{\circ}$. Then, since $ KM\parallel NP $, $ \angle MLO $ and $ \angle NOL $ are same - side interior angles, so they are supplementary. Wait, no, maybe I got the direction wrong. Wait, actually, $ \angle KLJ $ and $ \angle NOL $: Let's think about corresponding angles or alternate interior angles. Wait, $ KL $ and $ NO $ are parts of the parallel lines $ KM $ and $ NP $. $ LJ $ and $ OQ $ are parts of the transversal $ JQ $. So $ \angle KLJ $ and $ \angle NOL $: Let's see, $ \angle KLJ $ and $ \angle NOL $ are actually same - side interior angles? Wait, no, $ \angle KLJ $ and $ \angle NOL $: If we consider the parallel lines $ KM $ and $ NP $, and transversal $ JQ $, then $ \angle KLJ $ and $ \angle NOL $ are same - side interior angles? Wait, no, $ \angle KLJ $ is above $ KM $, and $ \angle NOL $ is above $ NP $? Wait, no, $ NO $ is part of $ NP $, going from $ N $ to $ P $, so $ NO $ is left - to - right, same as $ KM $. $ LJ $ is going up - right, $ OQ $ is going down - left. So $ \angle KLJ $ is the angle between $ KL $ (left - to - right) and $ LJ $ (up - right), and $ \angle NOL $ is the angle between $ NO $ (left - to - right) and $ OL $ (up - right, since $ OL $ is part of $ LJ $ extended? Wait, no, $ O $ is on $ NP $ and $ JQ $, so $ OL $ is the same line as $ LJ $, just extended. So $ \angle KLJ $ and $ \angle NOL $: Wait, actually, $ \angle KLJ $ and $ \angle NOL $ are alternate interior angles? No, alternate interior angles would be on opposite sides of the transversal. Wait, maybe I made a mistake. Let's recall that when two parallel lines are cut by a transversal, consecutive interior angles are supplementary. $ \angle KLJ $ and $ \angle MLO $ are vertical angles, so $ m\angle MLO = 130^{\circ}$. Then, $ KM\parallel NP $, so $ \angle MLO $ and $ \angle NOL $ are consecutive interior angles, so $ m\angle MLO+m\angle NOL = 180^{\circ}$. Wait, no, that would be if they are on the same side. Wait, maybe $ \angle KLJ $ and $ \angle NOL $ are actually same - side interior angles? Wait, no, let's look at the diagram again. $ KM $ is parallel to $ NP $, $ JQ $ is the transversal. $ \angle KLJ $ is at $ L $, between $ KL $ and $ LJ $. $ \angle NOL $ is at $ O $, between $ NO $ and $ OL $. Since $ KL \parallel NO $ (because $ KM \parallel NP $), and $ LJ $ is the transversal, then $ \angle KLJ $ and $ \angle NOL $ are same - side interior angles? Wait, no, same - side interior angles are on the same side of the transversal. Wait, $ KL $ and $ NO $ are parallel, $ LJ $ is the transversal. So $ \angle KLJ $ and $ \angle NOL $: $ KL $ is above $ NO $? No, $ KM $ and $ NP $ are two parallel lines, $ KM $ is the upper one, $ NP $ is the lower one. So $ KL $ is part of $ KM $, going left from $ L $, $ NO $ is part of $ NP $, going left from $ O $. $ LJ $ is going up - right from $ L $, $ OQ $ is going down - left from $ O $, so $ OL $ is going up - right from $ O $, same direction as $ LJ $. So $ \angle KLJ $ is the angle between $ KL $ (left) and $ LJ $ (up - right), and $ \angle NOL $ is the angle between $ NO $ (left) and $ OL $ (up - right). So these two angles are actually corresponding angles? Wait, corresponding angles are equal. But $ \angle KLJ $ is $ 130^{\circ}$, that can't be. Wait, no, maybe $ \angle KLJ $ and $ \angle NOL $ are supplementary. Wait, let's use the linear pair or supplementary angles. Wait, $ \angle KLJ $ and $ \angle MLJ $ are supplementary, because they form a linear pair. So $ m\angle KLJ + m\angle MLJ=180^{\circ}$, so $ m\angle MLJ = 180 - 130=50^{\circ}$. Then, since $ KM \parallel NP $, $ \angle MLJ $ and $ \angle NOL $ are corresponding angles, so $ m\angle NOL=m\angle MLJ = 50^{\circ}$. Ah, that makes sense. So first, find the supplementary angle of $ \angle KLJ $ (since they form a linear pair), then use the corresponding angles postulate (since $ KM \parallel NP $) to find $ \angle NOL $.

## Step1: Find the supplementary angle of $ \angle KLJ $
$ \angle KLJ $ and $ \angle MLJ $ form a linear pair, so they are supplementary. The sum of supplementary angles is $ 180^{\circ}$.
Given $ m\angle KLJ = 130^{\circ}$, then $ m\angle MLJ=180^{\circ}-m\angle KLJ $
$ m\angle MLJ = 180 - 130=50^{\circ}$

## Step2: Use corresponding angles postulate
Since $ KM\parallel NP $ and $ JQ $ is a transversal, $ \angle MLJ $ and $ \angle NOL $ are corresponding angles. Corresponding angles are equal when two parallel lines are cut by a transversal.
So $ m\angle NOL=m\angle MLJ = 50^{\circ}$

# Answer:
$ 50 $

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