Question

fh and ik are parallel lines.
which angles are alternate exterior angles?
∠ijl and ∠fgj
∠ijl and ∠kjg
∠ijl and ∠ijg
∠ijl and ∠hge

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: $\angle IJL$ and $\angle FGJ$: $\angle IJL$ is outside the parallel lines $FH$ and $IK$ (below $IK$), and $\angle FGJ$ is outside $FH$ (above $FH$), on opposite sides of transversal $EL$. Check their positions: transversal $EL$ cuts $FH$ (at $G$) and $IK$ (at $J$). $\angle IJL$ is at $J$ (exterior to $IK$ and $FH$), $\angle FGJ$ is at $G$ (exterior to $FH$ and $IK$), opposite sides of transversal. Wait, no, let's re - check. Wait, $\angle HGE$ and $\angle IJL$? Wait, no, let's look at the last option: $\angle IJL$ and $\angle HGE$. Wait, no, let's re - evaluate.

Wait, the transversal is line $EL$ (with points $E, G, J, L$). Parallel lines are $FH$ (with $F, G, H$) and $IK$ (with $I, J, K$).

Exterior angles: for line $FH$ and $IK$, exterior means outside the region between $FH$ and $IK$.

$\angle IJL$: at $J$, below $IK$, outside the "between" region.

$\angle HGE$: at $G$, above $FH$, outside the "between" region. And they are on opposite sides of transversal $EL$. Wait, no, the fourth option is $\angle IJL$ and $\angle HGE$. Wait, but let's check the first option again. Wait, maybe I made a mistake.

Wait, let's list the angles:

Transversal: $EL$ (passes through $G$ on $FH$ and $J$ on $IK$).

Parallel lines: $FH \parallel IK$.

Exterior angles:

- For $\angle IJL$: it is formed at $J$, between transversal $EL$ and line $IK$, outside the two parallel lines (below $IK$).

- For $\angle HGE$: formed at $G$, between transversal $EL$ and line $FH$, outside the two parallel lines (above $FH$), and on the opposite side of the transversal from $\angle IJL$.

Wait, but let's check the options again. The fourth option is $\angle IJL$ and $\angle HGE$. Wait, but in the options, the fourth option is $\angle IJL$ and $\angle HGE$. Wait, maybe my initial analysis was wrong. Wait, let's re - check the definition.

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

So, $\angle IJL$ is outside (below $IK$) and $\angle HGE$ is outside (above $FH$), and on opposite sides of transversal $EL$.

Wait, but let's check the first option: $\angle IJL$ and $\angle FGJ$. $\angle FGJ$: at $G$, between $FH$ and transversal $EL$, but on the $F$ side. Wait, $\angle FGJ$ is actually an interior angle? No, $\angle FGJ$: point $F - G - H$, transversal $EL$ at $G$. So $\angle FGJ$ is adjacent to $F$, so it's outside? Wait, no, the region between $FH$ and $IK$ is the middle. So above $FH$ and below $IK$ are exterior.

Wait, $\angle IJL$ is below $IK$ (exterior), $\angle HGE$ is above $FH$ (exterior), and they are on opposite sides of transversal $EL$. So the fourth option: $\angle IJL$ and $\angle HGE$ are alternate exterior angles.

Wait, but let's check the options again. The options are:

1. $\angle IJL$ and $\angle FGJ$

2. $\angle IJL$ and $\angle KJG$

3. $\angle IJL$ and $\angle IJG$

4. $\angle IJL$ and $\angle HGE$

Wait, $\angle KJG$ is adjacent to $K$, so it's an interior angle (between $FH$ and $IK$). $\angle IJG$ is also an interior angle (between $FH$ and $IK$). $\angle FGJ$: let's see, $F - G - H$, transversal $EL$ at $G$. $\angle FGJ$: from $F$ to $G$ to $J$? Wait, no, the angle at $G$: $\angle FGJ$ is the angle at $G$ between $F - G$ and $G - J$ (transversal). Wait, $F - G$ is part of $FH$, $G - J$ is part of transversal $EL$. So $\angle FGJ$ is outside? No, the area between $FH$ and $IK$ is between the two parallel lines. So above $FH$ and below $IK$ are exterior. So $\angle FGJ$: if $FH$ is horizontal, $F$ is left, $H$ is right. Transversal $EL$ goes from $E$ (top) to $L$ (bottom). So at $G$, $\angle FGJ$ is the angle between $FG$ (left along $FH$) and $GJ$ (down along transversal). So this angle is above $FH$? No, $FH$ is horizontal, $GJ$ is going down, so $\angle FGJ$ is below $FH$? Wait, I think I messed up the direction.

Let's use the correct definition: Alternate exterior angles are two angles that are outside the two parallel lines, and on opposite sides of the transversal.

So for parallel lines $FH$ and $IK$, cut by transversal $EL$ (with $G$ on $FH$ and $J$ on $IK$):

- Angles at $G$: $\angle FGE$ (above $FH$, left of transversal), $\angle HGE$ (above $FH$, right of transversal), $\angle FGL$ (below $FH$, left of transversal), $\angle HGL$ (below $FH$, right of transversal).

- Angles at $J$: $\angle IJE$ (above $IK$, left of transversal), $\angle KJE$ (above $IK$, right of transversal), $\angle IJL$ (below $IK$, left of transversal), $\angle KJL$ (below $IK$, right of transversal).

Now, alternate exterior angles: one at $G$ (outside $FH - IK$) and one at $J$ (outside $FH - IK$), on opposite sides of transversal.

So $\angle IJL$ (below $IK$, left of transversal) and $\angle HGE$ (above $FH$, right of transversal) are on opposite sides of transversal and outside the parallel lines.

Wait, but let's check the first option: $\angle IJL$ (below $IK$, left) and $\angle FGJ$ (which is $\angle FGL$? Wait, $\angle FGJ$: $F - G - J$, so it's $\angle FGL$ (since $J - G - L$ is a straight line? Wait, $E - G - J - L$ is a straight line, so $\angle FGJ$ is equal to $\angle FGL$ (vertical angles? No, $E - G - J - L$ is a straight line, so $\angle EGF$ and $\angle JGL$ are vertical angles, $\angle HGE$ and $\angle IJL$: wait, $\angle HGE$ is at $G$, above $FH$, right of transversal. $\angle IJL$ is at $J$, below $IK$, left of transversal. They are alternate exterior angles.

Wait, the fourth option is $\angle IJL$ and $\angle HGE$. So that's the correct pair.

Answer:

$\angle IJL$ and $\angle HGE$ (the fourth option: $\boldsymbol{\angle IJL}$ and $\boldsymbol{\angle HGE}$)