Question

$\overleftrightarrow{km}$ and $\overleftrightarrow{np}$ are parallel lines,

which angles are alternate exterior angles?

$\angle mlj$ and $\angle noq$ $\angle mlj$ and $\angle mlo$

$\angle mlj$ and $\angle klj$ $\angle mlj$ and $\angle nol$

Explanation:

Step1: Recall Alternate Exterior Angles Definition

Alternate exterior angles are formed when a transversal crosses two parallel lines. They lie outside the two parallel lines and on opposite sides of the transversal.

Step2: Analyze Each Option

- Option 1: $\angle MLJ$ and $\angle NOQ$: $\overleftrightarrow{KM} \parallel \overleftrightarrow{NP}$, transversal is $\overleftrightarrow{QJ}$. $\angle MLJ$ is outside $\overleftrightarrow{KM}$? No, $\angle MLJ$ is at the intersection of $\overleftrightarrow{QJ}$ and $\overleftrightarrow{KM}$, between the parallel lines? Wait, no, let's re - check. Wait, $\overleftrightarrow{KM}$ and $\overleftrightarrow{NP}$ are parallel, transversal $\overleftrightarrow{QJ}$. $\angle MLJ$: the lines are vertical (KM and NP) and transversal is slant (QJ). Wait, maybe I mis - identified. Wait, let's look at the positions. $\angle MLJ$: at L, between KM and the transversal? No, KM is vertical, QJ is slant. Wait, maybe the first option: $\angle MLJ$ and $\angle NOQ$. $\angle NOQ$ is at O, outside NP (since NP is vertical, O is on NP and QJ). $\angle MLJ$ is at L, outside KM? Wait, maybe I made a mistake. Wait, let's check the other options.
- Option 2: $\angle MLJ$ and $\angle MLO$: These two angles are adjacent, forming a linear pair? No, they are at the same vertex L, with a common side, so they are adjacent angles, not alternate exterior.
- Option 3: $\angle MLJ$ and $\angle KLJ$: These are adjacent angles, forming a linear pair (since KM is a straight line), so not alternate exterior.
- Wait, maybe I messed up the first option. Wait, let's re - define alternate exterior angles. When two parallel lines are cut by a transversal, alternate exterior angles are on the outside of the two parallel lines and on opposite sides of the transversal. So for parallel lines $\overleftrightarrow{KM}$ and $\overleftrightarrow{NP}$, transversal $\overleftrightarrow{QJ}$. $\angle MLJ$: let's see, KM is one parallel line, NP is the other. The exterior of the two parallel lines would be the regions not between KM and NP. $\angle MLJ$: at L, between KM and... Wait, no, KM is vertical, so the area between KM and NP is the region between the two vertical lines. So $\angle MLJ$: if we consider KM as a vertical line, and QJ as transversal, $\angle MLJ$ is on the "right" side of KM? And $\angle NOQ$ is on the "left" side of NP. Since KM and NP are parallel, and QJ is transversal, $\angle MLJ$ and $\angle NOQ$: $\angle MLJ$ is outside? Wait, maybe I was wrong earlier. Wait, let's check the other options again. The second option: $\angle MLJ$ and $\angle MLO$: same vertex, adjacent, so no. Third option: $\angle MLJ$ and $\angle KLJ$: linear pair, adjacent. Fourth option: $\angle MLJ$ and $\angle NOL$: $\angle NOL$ is at O, between NP and QJ? No. Wait, maybe the first option is correct. Wait, maybe my initial analysis was wrong. Let's re - do:

Alternate exterior angles: two angles that lie outside the two parallel lines, on opposite sides of the transversal.

For $\overleftrightarrow{KM} \parallel \overleftrightarrow{NP}$, transversal $\overleftrightarrow{QJ}$.

- $\angle MLJ$: at the intersection of $\overleftrightarrow{QJ}$ and $\overleftrightarrow{KM}$, outside the "between" region of $\overleftrightarrow{KM}$ and $\overleftrightarrow{NP}$? Wait, $\overleftrightarrow{KM}$ and $\overleftrightarrow{NP}$ are vertical, so the region between them is the area between the two vertical lines. $\angle MLJ$: if we look at the direction, $\overleftrightarrow{KM}$ is from K (top) to M (bottom), $\overleftrightarrow{NP}$ is from N (top) to P (bottom). The transversal $\overleftrightarrow{QJ}$ goes from Q (bottom - left) to J (top - right). So at point L (intersection of $\overleftrightarrow{QJ}$ and $\overleftrightarrow{KM}$), $\angle MLJ$ is the angle below the transversal and to the right of $\overleftrightarrow{KM}$. At point O (intersection of $\overleftrightarrow{QJ}$ and $\overleftrightarrow{NP}$), $\angle NOQ$ is the angle above the transversal and to the left of $\overleftrightarrow{NP}$. Wait, no, maybe I got the direction wrong. Wait, alternate exterior angles should be outside the two parallel lines and on opposite sides of the transversal. So if we consider the two parallel lines as $\overleftrightarrow{KM}$ and $\overleftrightarrow{NP}$ (vertical), and transversal $\overleftrightarrow{QJ}$ (slant), then the exterior regions are the regions not between $\overleftrightarrow{KM}$ and $\overleftrightarrow{NP}$. So $\angle MLJ$: is it outside? Wait, $\overleftrightarrow{KM}$ is one line, $\overleftrightarrow{NP}$ is the other. The area between them is between the two vertical lines. So $\angle MLJ$: at L, which is on $\overleftrightarrow{KM}$, and the angle $\angle MLJ$: if we extend our view, the "outside" of the two parallel lines would be to the left of $\overleftrightarrow{NP}$ and to the right of $\overleftrightarrow{KM}$? No, maybe the first option is correct. Let's check the other options again. The second option: $\angle MLJ$ and $\angle MLO$: same vertex, adjacent, so they are supplementary (linear pair), not alternate exterior. Third option: $\angle MLJ$ and $\angle KLJ$: same vertex, adjacent, linear pair. Fourth option: $\angle MLJ$ and $\angle NOL$: $\angle NOL$ is at O, between $\overleftrightarrow{NP}$ and the transversal? No. So the correct pair should be $\angle MLJ$ and $\angle NOQ$.

Answer:

$\angle MLJ$ and $\angle NOQ$ (the first option in the list of angle pairs)