select all the pairs of adjacent angles. ∠kli and ∠mli ∠jig and ∠hig ∠jil and ∠hil ∠jig and ∠jil

Question

Accepted Answer

# Explanation:
## Step1: Recall adjacent angles definition
Adjacent angles share a common side and vertex, and their non - common sides are opposite rays (or form a linear pair, or are adjacent in the angle structure).

## Step2: Analyze $ \angle KLI $ and $ \angle MLI $
- Vertex: Both angles have vertex $ L $.
- Common side: They share the side $ LI $.
- Non - common sides: $ LK $ and $ LM $ are opposite rays (since $ K - L - M $ is a straight line). So $ \angle KLI $ and $ \angle MLI $ are adjacent angles.

## Step3: Analyze $ \angle JIG $ and $ \angle HIG $
- Vertex: Both angles have vertex $ I $.
- Common side: They share the side $ IG $.
- Non - common sides: $ IJ $ and $ IH $ are opposite rays (since $ J - I - H $ is a straight line). So $ \angle JIG $ and $ \angle HIG $ are adjacent angles.

## Step4: Analyze $ \angle JIL $ and $ \angle HIL $
- Vertex: Both angles have vertex $ I $.
- Common side: They share the side $ IL $.
- Non - common sides: $ IJ $ and $ IH $ are opposite rays (since $ J - I - H $ is a straight line). So $ \angle JIL $ and $ \angle HIL $ are adjacent angles.

## Step5: Analyze $ \angle JIG $ and $ \angle JIL $
- Vertex: Both angles have vertex $ I $.
- Common side: They share the side $ IJ $.
- Non - common sides: $ IG $ and $ IL $ are not opposite rays (they form an angle between two different lines). But they still share a common vertex and a common side, and their non - common sides are distinct, so they are adjacent angles? Wait, no. Wait, $ \angle JIG $ has sides $ IJ $ and $ IG $, $ \angle JIL $ has sides $ IJ $ and $ IL $. The common vertex is $ I $, common side is $ IJ $, and the other sides $ IG $ and $ IL $ are different. But actually, let's re - check. Wait, in the diagram, $ IG $ and $ IL $ are two different lines intersecting at $ I $ with $ IJ $. Wait, maybe I made a mistake. Wait, adjacent angles must have the non - common sides forming a linear pair? No, the definition is: two angles are adjacent if they have a common side and a common vertex, and their interiors do not overlap. So $ \angle JIG $ and $ \angle JIL $: common vertex $ I $, common side $ IJ $, and the other sides $ IG $ and $ IL $ are different, and their interiors don't overlap. Wait, but in the diagram, $ IG $ and $ IL $ are two lines (the transversal and the other line). Wait, maybe the correct ones are the first three? Wait, no, let's re - examine the options.

Wait, the options are:

1. $ \angle KLI $ and $ \angle MLI $: adjacent (linear pair, share $ LI $, vertex $ L $, $ LK $ and $ LM $ are straight).

2. $ \angle JIG $ and $ \angle HIG $: adjacent (linear pair, share $ IG $, vertex $ I $, $ IJ $ and $ IH $ are straight).

3. $ \angle JIL $ and $ \angle HIL $: adjacent (linear pair, share $ IL $, vertex $ I $, $ IJ $ and $ IH $ are straight).

4. $ \angle JIG $ and $ \angle JIL $: Do they share a common side? $ \angle JIG $ has sides $ IJ $ and $ IG $, $ \angle JIL $ has sides $ IJ $ and $ IL $. So common side $ IJ $, common vertex $ I $, and non - common sides $ IG $ and $ IL $. But are their interiors overlapping? No. Wait, but in the diagram, $ IG $ and $ IL $ are two different lines (the transversal $ NG $ and the line $ HJ $ intersect at $ I $, and $ KL $ and $ HJ $ intersect at $ L $). Wait, maybe the fourth one is not adjacent. Wait, maybe I made a mistake. Let's go back to the definition: Adjacent angles are two angles that have a common vertex and a common side, and do not overlap.

For $ \angle KLI $ and $ \angle MLI $:

- Common vertex: $ L $
- Common side: $ LI $
- The other sides: $ LK $ and $ LM $, which are a straight line (so they form a linear pair, hence adjacent).

For $ \angle JIG $ and $ \angle HIG $:

- Common vertex: $ I $
- Common side: $ IG $
- The other sides: $ IJ $ and $ IH $, which are a straight line (linear pair, adjacent).

For $ \angle JIL $ and $ \angle HIL $:

- Common vertex: $ I $
- Common side: $ IL $
- The other sides: $ IJ $ and $ IH $, which are a straight line (linear pair, adjacent).

For $ \angle JIG $ and $ \angle JIL $:

- Common vertex: $ I $
- Common side: $ IJ $
- The other sides: $ IG $ and $ IL $. These two sides are not a straight line (since $ IG $ is part of transversal $ NG $ and $ IL $ is part of $ NG $? Wait, no, $ NG $ is a straight line, so $ IG $ and $ IL $ are on the same straight line? Wait, $ N - L - I - G $ is a straight line. Oh! Wait, $ NGL $ is a straight line. So $ IL $ and $ IG $ are on the same straight line. So $ \angle JIG $ and $ \angle JIL $: common vertex $ I $, common side $ IJ $, and the other sides $ IG $ and $ IL $ are on a straight line. So they form a linear pair? Wait, no, $ \angle JIG $ is between $ IJ $ and $ IG $, $ \angle JIL $ is between $ IJ $ and $ IL $. Since $ IG $ and $ IL $ are on a straight line, then $ \angle JIG+\angle JIL = 180^{\circ}$? Wait, no, because $ IJ $ is a different line. Wait, maybe I messed up the diagram. Let's assume that $ NGL $ is a straight line, $ KLM $ is a straight line, and $ HJI $ is a straight line.

So, $ \angle KLI $ and $ \angle MLI $: share $ LI $, vertex $ L $, $ LK $ and $ LM $ are straight: adjacent.

$ \angle JIG $ and $ \angle HIG $: share $ IG $, vertex $ I $, $ IJ $ and $ IH $ are straight: adjacent.

$ \angle JIL $ and $ \angle HIL $: share $ IL $, vertex $ I $, $ IJ $ and $ IH $ are straight: adjacent.

$ \angle JIG $ and $ \angle JIL $: share $ IJ $, vertex $ I $, $ IG $ and $ IL $ are straight (since $ NGL $ is straight), so they are adjacent (linear pair). Wait, but maybe the original problem's options are such that the correct ones are $ \angle KLI $ and $ \angle MLI $, $ \angle JIG $ and $ \angle HIG $, $ \angle JIL $ and $ \angle HIL $, and $ \angle JIG $ and $ \angle JIL $? But that seems too many. Wait, maybe the diagram is different. Alternatively, maybe the correct pairs are $ \angle KLI $ and $ \angle MLI $, $ \angle JIG $ and $ \angle HIG $, $ \angle JIL $ and $ \angle HIL $.

But according to the initial analysis, the pairs that are adjacent are:

- $ \angle KLI $ and $ \angle MLI $ (share vertex $ L $, common side $ LI $, linear pair)
- $ \angle JIG $ and $ \angle HIG $ (share vertex $ I $, common side $ IG $, linear pair)
- $ \angle JIL $ and $ \angle HIL $ (share vertex $ I $, common side $ IL $, linear pair)
- $ \angle JIG $ and $ \angle JIL $ (share vertex $ I $, common side $ IJ $, and $ IG $ and $ IL $ are on a straight line, so adjacent)

But maybe the problem's intended correct pairs are the first three or including the fourth. However, based on the definition of adjacent angles (common vertex, common side, non - overlapping interiors), all four? Wait, no, let's check again.

Wait, adjacent angles must have the two angles next to each other, sharing a side and vertex. So:

1. $ \angle KLI $ and $ \angle MLI $: Yes, adjacent (linear pair).

2. $ \angle JIG $ and $ \angle HIG $: Yes, adjacent (linear pair).

3. $ \angle JIL $ and $ \angle HIL $: Yes, adjacent (linear pair).

4. $ \angle JIG $ and $ \angle JIL $: Yes, because they share $ IJ $, vertex $ I $, and $ IG $ and $ IL $ are on the same line (so they are adjacent, forming a linear pair as well? Wait, no, $ \angle JIG $ is between $ IJ $ and $ IG $, $ \angle JIL $ is between $ IJ $ and $ IL $, and since $ IG $ and $ IL $ are colinear, the sum of these two angles is $ 180^{\circ}$, so they are adjacent (linear pair).

But maybe the problem's options are designed so that the correct ones are $ \angle KLI $ and $ \angle MLI $, $ \angle JIG $ and $ \angle HIG $, $ \angle JIL $ and $ \angle HIL $.

# Answer:
$ \angle KLI $ and $ \angle MLI $, $ \angle JIG $ and $ \angle HIG $, $ \angle JIL $ and $ \angle HIL $ (and possibly $ \angle JIG $ and $ \angle JIL $ depending on the diagram, but based on standard adjacent angle definitions, the first three are definite, and the fourth may also be correct)

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QUESTION IMAGE

Question

Explanation:

Step1: Recall adjacent angles definition

Step2: Analyze \( \angle KLI \) and \( \angle MLI \)

Step3: Analyze \( \angle JIG \) and \( \angle HIG \)

Step4: Analyze \( \angle JIL \) and \( \angle HIL \)

Step5: Analyze \( \angle JIG \) and \( \angle JIL \)

Answer: