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 an interior angle (or not exterior), so not alternate exterior.
- **∠JIG and ∠KLN**: ∠JIG is on the upper parallel line (exterior? No, ∠JIG is adjacent to the transversal on the upper line, but ∠KLN is on the lower line, outside? Wait, re - check. Wait, the two parallel lines are HJ and KM, transversal is GN. ∠JIG is on the upper line, right side of transversal; ∠KLN is on the lower line, left side of transversal, both exterior. Wait, no, let's re - define: Exterior angles are outside the two parallel lines (HJ and KM). So for transversal GN, the exterior angles are above HJ and below KM. ∠HIG is above HJ (exterior), ∠KLN is below KM (exterior), and they are on opposite sides of the transversal. Wait, maybe I made a mistake. Let's list all angles:
- For line HJ (upper) and KM (lower), transversal GN.
- Exterior angles: above HJ (∠HIG) and below KM (∠KLN, ∠MLN? No, ∠MLN is on the right of transversal, below KM, but ∠HIG is on the left of transversal, above HJ. Wait, alternate exterior angles should be on opposite sides of transversal and outside the two lines.
- ∠JIG and ∠KLN: ∠JIG is on the right of transversal, above HJ (interior? No, HJ is a line, so above HJ is exterior? Wait, the two lines are HJ and KM, so the region between HJ and KM is interior, outside is above HJ and below KM. So ∠JIG is on the right of transversal, above HJ (exterior? No, ∠JIG is between the transversal and the line HJ? Wait, no, HJ is a straight line, so the angles formed by transversal GN with HJ are ∠HIG (left, above HJ), ∠JIG (right, above HJ), ∠HIL (left, below HJ? No, HJ is a horizontal line, so above HJ and below HJ. Wait, maybe the two parallel lines are HJ (horizontal, left - right) and KM (horizontal, left - right), transversal is GN (slanted). So the exterior of the two lines (HJ and KM) is the area not between HJ and KM. So above HJ and below KM are exterior regions.
- ∠HIG: left of transversal, above HJ (exterior)
- ∠KLN: left of transversal? No, ∠KLN is on the lower line KM, left of transversal GN, below KM (exterior)
- ∠JIG: right of transversal, above HJ (exterior)
- ∠MLN: right of transversal, below KM (exterior)
- Now, alternate exterior angles: angles that are outside the two lines, on opposite sides of the transversal.
- So ∠HIG (left, above) and ∠KLN (left? No, that's same side. Wait, no, transversal is GN, so left and right of GN. ∠HIG is left of GN, ∠KLN is left of GN? No, ∠KLN is at point L, on line KM, with transversal GN. So the angle ∠KLN: the sides are LK (left along KM) and LN (down along GN). So ∠KLN is on the left of GN (since LN is going down - left from L). ∠HIG is on the left of GN (since GI is going up - right from I, so HIJ is horizontal, so ∠HIG is between HI (left along HJ) and GI (up - right), so left of GN. Wait, maybe I messed up. Let's use the correct definition: Alternate exterior angles are two angles that lie outside the two lines cut by the transversal, and on opposite sides of the transversal.
- So for lines HJ and KM (parallel), transversal GN.
- Outside the two lines: above HJ and below KM.
- Opposite sides of transversal: left and right of GN.
- So angle above HJ, left of GN: ∠HIG
- Angle below KM, right of GN: ∠MLN? No, ∠MLN is below KM, right of GN.
- Angle above HJ, right of GN: ∠JIG
- Angle below KM, left of GN: ∠KLN
- So alternate exterior angles should be ∠HIG (above, left) and ∠MLN (below, right)? No, the options are:
- ∠MLN and ∠HIG: ∠MLN is below KM, right of GN; ∠HIG is above HJ, left of GN. Are they alternate exterior? Let's check: outside the two lines (yes, above and below), opposite sides of transversal (left and right). So maybe. But let's check the other options.
- ∠JIG and ∠KLN: ∠JIG is above HJ, right of GN; ∠KLN is below KM, left of GN. Outside the two lines, opposite sides of transversal. So this is a pair.
- ∠HIG and ∠KLN: ∠HIG is above HJ, left of GN; ∠KLN is below KM, left of GN. Same side of transversal, so no.
- ∠HIG and ∠HIL: ∠HIL is adjacent to ∠HIG, same line, not alternate exterior.
- Wait, maybe I made a mistake in the first analysis. Let's re - learn: Alternate exterior angles are a pair of angles on the outer side of each of those two lines but on opposite sides of the transversal.
- So two parallel lines cut by a transversal.
- Line 1: HJ, Line 2: KM, Transversal: GN.
- Outer side of Line 1 (HJ): above HJ; Outer side of Line 2 (KM): below KM.
- Opposite sides of transversal: left and right.
- So angle on outer side of Line 1, left of transversal: ∠HIG
- Angle on outer side of Line 2, right of transversal: ∠MLN
- Angle on outer side of Line 1, right of transversal: ∠JIG
- Angle on outer side of Line 2, left of transversal: ∠KLN
- So alternate exterior angle pairs are (∠HIG, ∠MLN) and (∠JIG, ∠KLN)
- Now check the options:
- ∠MLN and ∠HIG: this is a pair.
- ∠JIG and ∠KLN: this is a pair.
- ∠HIG and ∠KLN: same side (left), so no.
- ∠HIG and ∠HIL: adjacent, no.
- But the options given:
- Option 1: ∠MLN and ∠HIG
- Option 2: ∠JIG and ∠KLN
- Option 3: ∠HIG and ∠KLN
- Option 4: ∠HIG and ∠HIL
- Wait, maybe the lines are HJ and KM, transversal GN. Let's look at the positions again. The angle ∠HIG: at point I, between HI (left) and GI (up - right). The angle ∠KLN: at point L, between KL (left) and LN (down - left). Wait, maybe the two parallel lines are HJ (HIJ) and KM (KLM), transversal is GLN (GILN). So the exterior angles: for HJ, the exterior is outside HIJ, so above HIJ (∠HIG) and below HIJ (but HIJ is a line, so below would be towards KM). Wait, I think I confused the direction. Let's use the standard definition with a diagram: 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 in the diagram, the two parallel lines are HJ (horizontal, left - right) and KM (horizontal, left - right), transversal is GN (slanted from bottom left to top right). So the angles:
- At point I (on HJ): ∠HIG (top - left of transversal), ∠JIG (top - right of transversal)
- At point L (on KM): ∠KLN (bottom - left of transversal), ∠MLN (bottom - right of transversal)
- Now, outside the two parallel lines (HJ and KM) means not between HJ and KM. So ∠HIG is above HJ (outside), ∠JIG is above HJ (inside? No, HJ is a line, so above and below. Wait, the region between HJ and KM is the interior, so above HJ and below KM are exterior.
- So ∠HIG (above HJ, left of transversal) and ∠MLN (below KM, right of transversal): outside, opposite sides.
- ∠JIG (above HJ, right of transversal) and ∠KLN (below KM, left of transversal): outside, opposite sides.
- So the pairs of alternate exterior angles are ∠MLN and ∠HIG, ∠JIG and ∠KLN. But let's check the options:
- Option 1: ∠MLN and ∠HIG – possible.
- Option 2: ∠JIG and ∠KLN – possible.
- Option 3: ∠HIG and ∠KLN – same side (left), so no.
- Option 4: ∠HIG and ∠HIL – adjacent, not alternate exterior.
- Wait, but maybe the answer is ∠JIG and ∠KLN and ∠MLN and ∠HIG? But let's check again. Wait, maybe I made a mistake in ∠MLN and ∠HIG. ∠MLN is at point L, on KM, below KM, right of transversal. ∠HIG is at point I, on HJ, above HJ, left of transversal. So they are outside the two lines, opposite sides of transversal. ∠JIG is at point I, on HJ, above HJ, right of transversal. ∠KLN is at point L, on KM, below KM, left of transversal. So they are also outside, opposite sides.
- But the options given: let's see the original options again:
- ∠MLN and ∠HIG
- ∠JIG and ∠KLN
- ∠HIG and ∠KLN
- ∠HIG and ∠HIL
- So the correct pairs are ∠MLN and ∠HIG, ∠JIG and ∠KLN. But maybe the answer is ∠JIG and ∠KLN (option 2) and ∠MLN and ∠HIG (option 1). But let's check with the definition again.
- Alternate exterior angles: two angles that are outside the two parallel lines, and on opposite sides of the transversal. So for the two parallel lines (HJ and KM) cut by transversal GN:
- Outside the lines: above HJ (∠HIG, ∠JIG) and below KM (∠KLN, ∠MLN)
- Opposite sides of transversal: left (∠HIG, ∠KLN) and right (∠JIG, ∠MLN)
- So alternate exterior angles are (∠HIG, ∠MLN) and (∠JIG, ∠KLN) because ∠HIG is left - outside, ∠MLN is right - outside; ∠JIG is right - outside, ∠KLN is left - outside.
- So the correct pairs are ∠MLN and ∠HIG, ∠JIG and ∠KLN.

Answer:

∠JIG and ∠KLN, ∠MLN and ∠HIG (But according to the options, if we have to choose from the given, the correct pairs are ∠JIG and ∠KLN (second option) and ∠MLN and ∠HIG (first option). But maybe the intended answer is ∠JIG and ∠KLN and ∠MLN and ∠HIG. However, let's re - check the diagram again. The two parallel lines are HJ (HIJ) and KM (KLM), transversal is GLN. The angle ∠HIG: top - left, ∠KLN: bottom - left, ∠JIG: top - right, ∠MLN: bottom - right. So alternate exterior angles are top - left and bottom - right (∠HIG and ∠MLN), top - right and bottom - left (∠JIG and ∠KLN). So the correct pairs are ∠MLN and ∠HIG, ∠JIG and ∠KLN.