Question

select all the pairs of alternate interior angles.
∠hil and ∠jig
∠kli and ∠jil
∠kli and ∠jig
∠mli and ∠hil

Explanation:

Step1: Recall Alternate Interior Angles Definition

Alternate interior angles are formed when a transversal crosses two parallel lines. They lie between the two lines (interior) and on opposite sides of the transversal.

Step2: Analyze Each Pair

• ∠HIL and ∠JIG: ∠JIG is vertical to another angle, not alternate interior with ∠HIL. Incorrect.
• ∠KLI and ∠JIL: ∠KLI and ∠JIL are between the two parallel lines (KM and HJ) and on opposite sides of transversal NG. Correct.
• ∠KLI and ∠JIG: ∠JIG is not between the lines. Incorrect.
• ∠MLI and ∠HIL: ∠MLI and ∠HIL are between the two parallel lines and on opposite sides of transversal NG. Correct. Wait, re - evaluating: Wait, initial wrong analysis. Let's re - check the lines. The two parallel 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).

Alternate interior angles: For transversal NG cutting KM and HJ:

• ∠KLI (on KM, above NG? Wait, no, between KM and HJ. Wait, KM and HJ are parallel, transversal NG. So interior angles are between KM and HJ. ∠KLI is at L, between KM and NG? Wait, no, KM and HJ are the two parallel lines. So the region between KM and HJ is the interior. So ∠KLI: at L, between KM and NG? No, KM is K - L - M (vertical line), HJ is H - I - J (vertical line). Transversal NG is a slant line. So the interior of the two parallel lines (KM and HJ) is between them. So ∠KLI: at L, between KM and HJ? Wait, KM is K (top) - L - M (bottom), HJ is H (top) - I - J (bottom). So the area between KM and HJ is between the two vertical lines. The transversal NG crosses them at L and I. So alternate interior angles should be: at L (on KM) and I (on HJ), between the two vertical lines, and on opposite sides of NG.

So ∠KLI: at L, between KM and NG (but KM is vertical, NG is slant). Wait, maybe better to use the definition: alternate interior angles are non - adjacent, interior (between the two lines), and on alternate sides of the transversal.

So for transversal NG, cutting parallel lines KM and HJ:

• ∠KLI (at L, on KM, inside the two lines) and ∠JIL (at I, on HJ, inside the two lines, opposite side of transversal) – this is correct.
• ∠MLI (at L, on KM, inside the two lines) and ∠HIL (at I, on HJ, inside the two lines, opposite side of transversal) – this is correct.

Wait, the initial checked options were wrong. Let's re - do:

Correct pairs:

• ∠KLI and ∠JIL: ∠KLI is at L, between KM and HJ (interior), ∠JIL is at I, between KM and HJ (interior), and on opposite sides of NG (transversal). Correct.
• ∠MLI and ∠HIL: ∠MLI is at L, between KM and HJ (interior), ∠HIL is at I, between KM and HJ (interior), and on opposite sides of NG (transversal). Correct.

The other pairs:

• ∠HIL and ∠JIG: ∠JIG is vertical to ∠NIL, not alternate interior.
• ∠KLI and ∠JIG: ∠JIG is outside the two lines (HJ and KM), so not interior.

So the correct pairs are ∠KLI and ∠JIL, ∠MLI and ∠HIL.

Answer:

The correct pairs of alternate interior angles are $\angle KLI$ and $\angle JIL$, $\angle MLI$ and $\angle HIL$. So the correct options (from the given choices) are the ones with $\angle KLI$ and $\angle JIL$, and $\angle MLI$ and $\angle HIL$ (assuming the last option is $\angle MLI$ and $\angle HIL$ as per the image's last checkboxed option, maybe a mis - check initially). So if the options are:

• $\angle HIL$ and $\angle JIG$: Incorrect
• $\angle KLI$ and $\angle JIL$: Correct
• $\angle KLI$ and $\angle JIG$: Incorrect
• $\angle MLI$ and $\angle HIL$: Correct

So the answer is the options with $\angle KLI$ and $\angle JIL$, and $\angle MLI$ and $\angle HIL$ (the second and fourth options in the given layout, assuming the fourth is $\angle MLI$ and $\angle HIL$).