Question

select all the pairs of alternate exterior angles.
∠kli and ∠mln
∠hil and ∠mli
∠jil and ∠mli
∠mln and ∠hig

Explanation:

Step1: Recall Alternate Exterior Angles Definition

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

Step2: Analyze Each Pair

- **∠KLI and ∠MLN**: ∠KLI is outside the two vertical lines (KM and HJ) and on one side of transversal NG. ∠MLN is outside and on the opposite side? Wait, no, re - check. Wait, actually, let's look at the lines. The two lines are KM (with points K, L, M) and HJ (with points H, I, J). The transversal is NG (with points N, L, I, G).
- ∠KLI: outside the region between KM and HJ, on the upper - left of transversal NG.
- ∠MLN: outside the region between KM and HJ, on the lower - left of transversal NG. Wait, no, maybe I made a mistake. Wait, alternate exterior angles should be on different sides of the transversal and outside the two lines. Let's re - evaluate.
- **∠HIL and ∠MLI**: ∠HIL is outside the two lines (KM and HJ) on the upper - right of transversal NG, ∠MLI is outside on the lower - left? No, wait, maybe the correct pairs are:
- The correct alternate exterior angle pairs:
- ∠MLN and ∠HIG: ∠MLN is outside the two lines (KM and HJ) on the left - side of transversal NG, ∠HIG is outside on the right - side of transversal NG.
- ∠KLI and ∠JIL: Wait, no, let's use the definition properly. When two lines are cut by a transversal, alternate exterior angles are non - adjacent, outside the two lines, and on opposite sides of the transversal.
- Let's list the angles:
- For lines KM and HJ, transversal NG.
- Exterior angles: ∠KLI (above KM, left of NG), ∠MLN (below KM, left of NG) – no, that's not alternate. Wait, ∠KLI (outside, left of NG) and ∠JIL (outside, right of NG)? No, ∠JIL is below HJ, right of NG. Wait, maybe the correct pairs are ∠MLN and ∠HIG, and ∠KLI and ∠JIL? But the given options:
- Option 1: ∠KLI and ∠MLN – these are adjacent? No, ∠KLI and ∠MLN: ∠KLI is at L, between KM and NG, ∠MLN is at L, between KM and NG (wait, no, KM is a straight line, L is on KM. So ∠KLI and ∠MLN: ∠KLI + ∠MLN = 180°? No, maybe I misread the diagram.
- Wait, the correct alternate exterior angle pairs should be:
- ∠MLN and ∠HIG: ∠MLN is outside the two parallel - like lines (KM and HJ) on the left of transversal NG, ∠HIG is outside on the right of transversal NG.
- ∠KLI and ∠JIL: ∠KLI is outside on the left - upper of transversal NG, ∠JIL is outside on the right - lower of transversal NG. But in the given options, the correct pairs are ∠MLN and ∠HIG, and let's check the options again. The options are:
- ∠KLI and ∠MLN: No, they are adjacent supplementary?
- ∠HIL and ∠MLI: ∠HIL is at I, ∠MLI is at L – no.
- ∠JIL and ∠MLI: No.
- ∠MLN and ∠HIG: Yes, because ∠MLN is outside the two lines (KM and HJ) on the left of transversal NG, ∠HIG is outside on the right of transversal NG. Also, ∠KLI and ∠JIL? But in the given options, the correct pairs are ∠MLN and ∠HIG, and maybe ∠KLI and ∠JIL, but the options given:
Wait, maybe the initial selection was wrong. Let's start over.

Alternate exterior angles: two angles that lie outside the two lines cut by a transversal, and on opposite sides of the transversal.

Lines: Let's assume KM and HJ are the two lines (parallel or not), and NG is the transversal.

- Angles outside KM and HJ:
- Above KM: ∠KLI
- Below KM: ∠MLN
- Above HJ: ∠HIL
- Below HJ: ∠JIL

Transversal NG: divides the plane into left and right.

- Left of NG: ∠KLI, ∠MLN
- Right of NG: ∠HIL, ∠JIL

So alternate exterior angles should be one from left - outside and one from right - outside, and not adjacent.

So ∠KLI (left - upper - outside) and ∠JIL (right - lower - outside) – no, vertical? Wait, no. ∠MLN (left - lower - outside) and ∠HIL (right - upper - outside) – yes! ∠MLN is left - lower - outside, ∠HIL is right - upper - outside. Wait, but in the options, we have ∠MLN and ∠HIG. ∠HIG is equal to ∠HIL (vertical angles). So ∠MLN and ∠HIG are alternate exterior angles because ∠HIG is vertical to ∠HIL, so ∠MLN and ∠HIG are alternate exterior.

Also, ∠KLI and ∠JIL? No, ∠JIL is below HJ, right of NG. ∠KLI is above KM, left of NG. They are alternate exterior. But in the given options, the correct pairs are:

The correct pairs of alternate exterior angles are ∠MLN and ∠HIG, and ∠KLI and ∠JIL (but ∠JIL is not in the options as a pair with ∠KLI). Wait, the options are:

1. ∠KLI and ∠MLN
2. ∠HIL and ∠MLI
3. ∠JIL and ∠MLI
4. ∠MLN and ∠HIG

Let's check ∠MLN and ∠HIG:

- ∠MLN: outside the two lines (KM and HJ), on the left side of transversal NG.
- ∠HIG: outside the two lines (KM and HJ), on the right side of transversal NG. And they are on opposite sides of the transversal. So this is a pair of alternate exterior angles.

∠KLI and ∠MLN: These two angles are adjacent (they form a linear pair? Wait, ∠KLI + ∠MLN = 180°? Because L is on KM, so ∠KLI and ∠MLN are supplementary (linear pair), not alternate exterior.

∠HIL and ∠MLI: ∠HIL is at I, ∠MLI is at L. They are not alternate exterior.

∠JIL and ∠MLI: ∠JIL is at I, ∠MLI is at L. Not alternate exterior.

So the correct pair is ∠MLN and ∠HIG.

Answer:

The correct pair of alternate exterior angles is $\angle MLN$ and $\angle HIG$ (among the given options, the pair $\angle MLN$ and $\angle HIG$ is the correct one for alternate exterior angles). If we consider the options, the correct option is the one with $\angle MLN$ and $\angle HIG$.