Question

select all the pairs of alternate exterior angles.
∠mln and ∠hig ∠jig and ∠kln
∠hig and ∠kln ∠hig and ∠hil

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

- **∠MLN and ∠HIG**: ∠MLN is inside the two lines (between HJ and KM), not exterior. Eliminate.
- **∠JIG and ∠KLN**: ∠JIG is on the right of transversal GL, ∠KLN is on the left of transversal GL, both outside the two lines (HJ and KM). This is a pair of alternate exterior angles.
- **∠HIG and ∠KLN**: ∠HIG is on the left of transversal GL, ∠KLN is also on the left. Same side, not alternate. Eliminate.
- **∠HIG and ∠HIL**: These are adjacent angles, not formed by two parallel lines and a transversal in the alternate exterior way. Eliminate. Wait, recheck: Wait, transversal is GL, lines HJ and KM. ∠JIG (right of GL, outside) and ∠KLN (left of GL, outside) – yes. Also, wait another pair? Wait, maybe I missed. Wait, ∠HIG: HJ is top line, KM is bottom line. Transversal GL. ∠HIG is outside (above HJ, left of GL), ∠KLN is outside (below KM, left of GL)? No, wait KM: K is left, M is right. So ∠KLN: at L, K is left, N is down. So ∠KLN is outside (left of KM, below). ∠JIG: at I, J is right, G is up. So ∠JIG is outside (right of HJ, above). Wait, maybe another pair: ∠HIG (left of GL, above HJ) and ∠MLN? No, ∠MLN is right of GL, below KM. Wait, no. Wait the correct pairs: when two parallel lines (HJ || KM) cut by transversal GL, alternate exterior angles are on opposite sides of transversal, outside the two lines. So ∠JIG (exterior, right of GL, above HJ) and ∠KLN (exterior, left of GL, below KM) – opposite sides of GL, outside. Also, ∠HIG (exterior, left of GL, above HJ) and ∠MLN (exterior, right of GL, below KM)? Wait the options: the first option is ∠MLN and ∠HIG. Let's check ∠MLN: at L, M is right, N is down. So ∠MLN is below KM, right of GL. ∠HIG is above HJ, left of GL. So they are on opposite sides of GL, outside the two lines (HJ and KM). So ∠MLN and ∠HIG: yes, that's also a pair. Wait I made a mistake earlier. Let's re-express:

Two parallel lines: HJ (H to J) and KM (K to M). Transversal: GL (G to N).

Exterior angles: angles outside the region between HJ and KM.

For ∠MLN: at L, between KM and GL, but on the outside (below KM, right of GL).

∠HIG: at I, between HJ and GL, on the outside (above HJ, left of GL).

So ∠MLN (outside, right of GL, below KM) and ∠HIG (outside, left of GL, above HJ) – alternate exterior (opposite sides of GL, outside the two lines).

∠JIG: at I, outside (above HJ, right of GL), ∠KLN: at L, outside (below KM, left of GL) – opposite sides of GL, outside. So two pairs? Wait the options:

First option: ∠MLN and ∠HIG – yes, alternate exterior.

Second option: ∠JIG and ∠KLN – yes, alternate exterior.

Wait my initial analysis was wrong. Let's redo:

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

Lines: HJ (horizontal, H left, J right) and KM (horizontal, K left, M right) – so HJ || KM.

Transversal: GL (diagonal, G up, N down).

Angles:

- ∠HIG: vertex I, sides HI (left along HJ) and IG (up along GL). So outside the "between" HJ and KM (above HJ), left of transversal GL.

- ∠JIG: vertex I, sides JI (right along HJ) and IG (up along GL). Outside (above HJ), right of transversal GL.

- ∠KLN: vertex L, sides KL (left along KM) and LN (down along GL). Outside (below KM), left of transversal GL.

- ∠MLN: vertex L, sides ML (right along KM) and LN (down along GL). Outside (below KM), right of transversal GL.

So alternate exterior pairs:

- ∠HIG (left of GL, above HJ) and ∠MLN (right of GL, below KM) – opposite sides of GL, outside.

- ∠JIG (right of GL, above HJ) and ∠KLN (left of GL, below KM) – opposite sides of GL, outside.

Now check the options:

1. ∠MLN and ∠HIG – yes, this is a pair.

2. ∠JIG and ∠KLN – yes, this is a pair.

3. ∠HIG and ∠KLN – both left of GL, same side – no.

4. ∠HIG and ∠HIL – adjacent, not alternate exterior – no.

Wait but the options given: the first option is ∠MLN and ∠HIG, second is ∠JIG and ∠KLN, third is ∠HIG and ∠KLN, fourth is ∠HIG and ∠HIL.

So the correct pairs are ∠MLN and ∠HIG, and ∠JIG and ∠KLN.

Answer:

The pairs of alternate exterior angles are ∠MLN and ∠HIG, ∠JIG and ∠KLN. (So select these two options: "∠MLN and ∠HIG" and "∠JIG and ∠KLN")