select all the pairs of alternate exterior angles.
∠ijl and ∠hgj ∠kjg and ∠hge
∠fge and ∠kjl ∠hge and ∠ijl

Question

select all the pairs of alternate exterior angles.
∠ijl and ∠hgj  ∠kjg and ∠hge
∠fge and ∠kjl  ∠hge and ∠ijl

Accepted Answer

# Explanation:
## Step1: Recall Alternate Exterior Angles Definition
Alternate exterior angles are formed when a transversal crosses two parallel lines. They lie outside the two lines and on opposite sides of the transversal.

## Step2: Analyze Each Pair
- **∠IJL and ∠HGJ**: ∠IJL is outside the lines $FH$ and $IK$, ∠HGJ is outside too, opposite sides of transversal $EL$. Valid.
- **∠KJG and ∠HGE**: ∠KJG is outside, ∠HGE is outside, opposite sides of transversal. Valid.
- **∠FGE and ∠KJL**: ∠FGE is outside, ∠KJL is outside, opposite sides of transversal. Valid.
- **∠HGE and ∠IJL**: ∠HGE is outside, ∠IJL is outside, opposite sides of transversal. Valid. Wait, no—wait, let's re - check. Wait, maybe I made a mistake earlier. Wait, alternate exterior angles: the two lines are $FH$ (with points $F, G, H$) and $IK$ (with points $I, J, K$), transversal is $EL$ (with points $E, G, J, L$).

So for $FH$ and $IK$ cut by $EL$:

- Exterior angles: above $FH$ and below $IK$, or below $FH$ and above $IK$, on opposite sides of $EL$.

Let's re - evaluate each pair:

1. **∠IJL and ∠HGJ**: ∠IJL is below $IK$ (exterior), ∠HGJ is above $FH$ (exterior), opposite sides of $EL$. Valid.
2. **∠KJG and ∠HGE**: ∠KJG is above $IK$ (exterior), ∠HGE is below $FH$ (exterior), opposite sides of $EL$. Valid.
3. **∠FGE and ∠KJL**: ∠FGE is below $FH$ (exterior), ∠KJL is above $IK$ (exterior), opposite sides of $EL$. Valid.
4. **∠HGE and ∠IJL**: ∠HGE is below $FH$ (exterior), ∠IJL is below $IK$ (exterior) — same side? No, wait, transversal $EL$: ∠HGE is on one side of $EL$ (right of $EL$ above $FH$? Wait, maybe my initial analysis was wrong. Wait, let's use the correct definition: alternate exterior angles are non - adjacent, outside the two parallel lines, and on opposite sides of the transversal.

Wait, maybe the correct pairs are:

- ∠FGE (outside $FH$, left of $EL$) and ∠KJL (outside $IK$, right of $EL$) — alternate exterior.
- ∠HGE (outside $FH$, right of $EL$) and ∠IJL (outside $IK$, left of $EL$) — alternate exterior.
- ∠KJG (outside $IK$, right of $EL$) and ∠HGE (outside $FH$, right of $EL$) — no, same side. Wait, no, ∠KJG is above $IK$, ∠HGE is below $FH$, opposite sides of transversal. Wait, maybe the original checked pairs: let's see the diagram again.

Wait, the lines $FH$ (horizontal, $F$ left, $H$ right) and $IK$ (horizontal, $I$ left, $K$ right), transversal $EL$ (diagonal, $E$ top - left, $L$ bottom - right).

So exterior angles:

- For $FH$: angles above $FH$ (∠HGE) and below $FH$ (∠FGE).
- For $IK$: angles above $IK$ (∠KJG) and below $IK$ (∠IJL).

Alternate exterior angles:

- ∠FGE (below $FH$, left of $EL$) and ∠KJL (below $IK$, right of $EL$)? No, ∠KJL is below $IK$? Wait, ∠KJL: point $J$, $K$ is right, $L$ is down, so ∠KJL is below $IK$, right of $EL$. ∠FGE is below $FH$, left of $EL$. So they are alternate exterior.

- ∠HGE (above $FH$, right of $EL$) and ∠IJL (below $IK$, left of $EL$)? Wait, ∠IJL is below $IK$, left of $EL$. ∠HGE is above $FH$, right of $EL$. So they are alternate exterior.

- ∠KJG (above $IK$, right of $EL$) and ∠HGE (above $FH$, right of $EL$) — no, same side. Wait, no, ∠KJG is above $IK$, ∠HGE is above $FH$, opposite sides of transversal? Wait, transversal $EL$: ∠KJG is on the right side of $EL$ (above $IK$), ∠HGE is on the right side of $EL$ (above $FH$) — same side. So that pair is wrong.

- ∠IJL (below $IK$, left of $EL$) and ∠HGJ: ∠HGJ is at $G$, between $FH$ and $EL$ — interior? No, ∠HGJ is between $FH$ and $EL$, so interior. So that pair is wrong.

- ∠HGE (above $FH$, right of $EL$) and ∠IJL (below $IK$, left of $EL$): alternate exterior.

- ∠FGE (below $FH$, left of $EL$) and ∠KJL (below $IK$, right of $EL$): alternate exterior.

- ∠KJG (above $IK$, right of $EL$) and ∠HGE (above $FH$, right of $EL$): same side, not alternate.

- ∠IJL (below $IK$, left of $EL$) and ∠HGJ: interior, not exterior.

So the correct pairs should be ∠FGE and ∠KJL, ∠HGE and ∠IJL. Wait, but the original has some checked. Maybe I misread the diagram.

Wait, let's re - define alternate exterior angles properly: When two parallel lines are cut by a transversal, alternate exterior angles are congruent and lie outside the two lines, on opposite sides of the transversal.

So lines $FH$ and $IK$ (assumed parallel), transversal $EL$.

- Angles outside $FH$ and $IK$:

- ∠FGE (outside $FH$, below $FH$, left of $EL$)

- ∠HGE (outside $FH$, above $FH$, right of $EL$)

- ∠IJL (outside $IK$, below $IK$, left of $EL$)

- ∠KJG (outside $IK$, above $IK$, right of $EL$)

Now, alternate exterior angles:

- ∠FGE (left - below - outside) and ∠KJG (right - above - outside)? No, different sides. Wait, no: opposite sides of transversal (EL) means one on left of EL, one on right of EL.

So:

- Left of EL: ∠FGE (below FH), ∠IJL (below IK)

- Right of EL: ∠HGE (above FH), ∠KJG (above IK)

Alternate exterior:

- ∠FGE (left - below - outside) and ∠KJG (right - above - outside)? No, that's not alternate. Wait, I think I messed up. The correct definition is: two angles are alternate exterior angles if they are on the exterior of the two lines, and on alternate (opposite) sides of the transversal.

So for two lines $a$ and $b$, transversal $t$:

- Exterior: not between $a$ and $b$

- Alternate sides: one on left of $t$, one on right of $t$

So for $a = FH$, $b = IK$, $t = EL$:

- Exterior angles:

- On $a$ (FH): ∠HGE (above FH, right of EL), ∠FGE (below FH, left of EL)

- On $b$ (IK): ∠KJG (above IK, right of EL), ∠IJL (below IK, left of EL)

Now, alternate exterior pairs:

- ∠FGE (left of EL, below FH) and ∠KJG (right of EL, above IK): no, different vertical positions. Wait, no, the vertical position doesn't matter. Wait, no, the two lines are horizontal, so "exterior" is above the top line and below the bottom line? No, exterior is outside the region between the two lines. So if the two lines are horizontal (FH above IK), then exterior is above FH and below IK.

Ah! That's the mistake. If $FH$ is above $IK$ (since $G$ is above $J$), then:

- Exterior to the region between $FH$ and $IK$:

- Above $FH$: ∠HGE

- Below $IK$: ∠IJL

And also, on the other side of the transversal:

- Below $FH$: ∠FGE

- Above $IK$: ∠KJG

Now, alternate exterior angles:

- ∠HGE (above $FH$, right of $EL$) and ∠IJL (below $IK$, left of $EL$): alternate (opposite sides of $EL$) and exterior (outside the region between $FH$ and $IK$).

- ∠FGE (below $FH$, left of $EL$) and ∠KJG (above $IK$, right of $EL$): alternate (opposite sides of $EL$) and exterior (outside the region between $FH$ and $IK$).

So the correct pairs are:

- ∠FGE and ∠KJL (wait, ∠KJL is the same as ∠KJG? No, ∠KJL: point $J$, $K$ is right, $L$ is down, so ∠KJL is below $IK$, right of $EL$. Wait, no, if $IK$ is above $JL$ (since $J$ is on $IK$ and $L$ is below $J$), then below $IK$ is ∠IJL (left of $EL$) and ∠KJL (right of $EL$).

So re - correcting:

Lines $FH$ (top horizontal) and $IK$ (bottom horizontal), transversal $EL$ (diagonal from $E$ (top - left) to $L$ (bottom - right)).

Exterior angles (outside the space between $FH$ and $IK$):

- Above $FH$: ∠HGE (right of $EL$)

- Below $IK$: ∠IJL (left of $EL$), ∠KJL (right of $EL$)

- Below $FH$: ∠FGE (left of $EL$)

- Above $IK$: ∠KJG (right of $EL$)

Now, alternate exterior angles (outside the two lines, opposite sides of transversal):

- ∠FGE (below $FH$, left of $EL$) and ∠KJL (below $IK$, right of $EL$): alternate (left - right) and exterior (below top line, below bottom line? No, exterior is outside the region between the two lines. So below $FH$ (top line) and below $IK$ (bottom line) is outside the region between them. So yes, exterior. And opposite sides of transversal. So this is a valid pair.

- ∠HGE (above $FH$, right of $EL$) and ∠IJL (below $IK$, left of $EL$): alternate (right - left) and exterior (above top line, below bottom line), outside the region between them. Valid pair.

- ∠KJG (above $IK$, right of $EL$) and ∠FGE (below $FH$, left of $EL$): alternate (right - left) and exterior (above bottom line, below top line), outside the region between them. Wait, above $IK$ (bottom line) and below $FH$ (top line) is outside the region between them. So valid.

- ∠HGE (above $FH$, right of $EL$) and ∠IJL (below $IK$, left of $EL$): we already said this is valid.

Wait, the original options:

1. ∠IJL and ∠HGJ: ∠HGJ is between $FH$ and $EL$, so interior. Not exterior. So this is wrong.

2. ∠KJG and ∠HGE: ∠KJG is above $IK$ (exterior), ∠HGE is above $FH$ (exterior), same side of transversal (right of $EL$). So not alternate. Wrong.

3. ∠FGE and ∠KJL: ∠FGE is below $FH$ (exterior, left of $EL$), ∠KJL is below $IK$ (exterior, right of $EL$). Alternate sides of transversal. Valid.

4. ∠HGE and ∠IJL: ∠HGE is above $FH$ (exterior, right of $EL$), ∠IJL is below $IK$ (exterior, left of $EL$). Alternate sides of transversal. Valid.

So the correct pairs are ∠FGE and ∠KJL, ∠HGE and ∠IJL.

But the original checked pairs: maybe the diagram has $FH$ and $IK$ as parallel, and the correct alternate exterior angle pairs are ∠FGE and ∠KJL, ∠HGE and ∠IJL, and also let's re - check the options:

- ∠IJL and ∠HGJ: ∠HGJ is interior (between $FH$ and $EL$), so no.

- ∠KJG and ∠HGE: same side of transversal, so no.

- ∠FGE and ∠KJL: yes.

- ∠HGE and ∠IJL: yes.

So the correct pairs are ∠FGE and ∠KJL, ∠HGE and ∠IJL.

# Answer:
The correct pairs of alternate exterior angles are $\angle FGE$ and $\angle KJL$, $\angle HGE$ and $\angle IJL$ (and also, if we consider the other valid one, $\angle FGE$ and $\angle KJG$ but based on the options, $\angle FGE$ and $\angle KJL$, $\angle HGE$ and $\angle IJL$ are the correct ones from the given options. Wait, in the options, the pairs are:

- $\angle IJL$ and $\angle HGJ$ (wrong)

- $\angle KJG$ and $\angle HGE$ (wrong)

- $\angle FGE$ and $\angle KJL$ (correct)

- $\angle HGE$ and $\angle IJL$ (correct)

So the correct options (the ones with checkmarks that are correct) are $\boldsymbol{\angle FGE}$ and $\boldsymbol{\angle KJL}$, $\boldsymbol{\angle HGE}$ and $\boldsymbol{\angle IJL}$ (i.e., the third and fourth options in the given set of options).

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QUESTION IMAGE

Question

Explanation:

Step1: Recall Alternate Exterior Angles Definition

Step2: Analyze Each Pair

Answer: