Question

select all the pairs of adjacent angles.
∠kjl and ∠ijl ∠hge and ∠fge
∠fgj and ∠fge ∠ijl and ∠ijg

Explanation:

Step1: Recall adjacent angles definition

Adjacent angles share a common side and vertex, and their non - common sides are on opposite sides of the common side.

Step2: Analyze ∠KJL and ∠IJL

∠KJL and ∠IJL share the common side JL and the common vertex J. Their non - common sides JK and JI are on a straight line (since IJK is a straight line), so they are adjacent angles.

Step3: Analyze ∠HGE and ∠FGE

∠HGE and ∠FGE share the common side GE and the common vertex G. Their non - common sides GH and GF are on a straight line (since FGH is a straight line), so they are adjacent angles.

Step4: Analyze ∠FGJ and ∠FGE

∠FGJ and ∠FGE share the common side FG and the common vertex G. Their non - common sides GJ and GE are not on a straight line? Wait, no, let's check again. Wait, actually, ∠FGJ and ∠FGE: the common vertex is G, common side is FG. The other sides are GJ and GE. Wait, but in the diagram, lines EGL and FGH are two lines intersecting at G, and line IJK is parallel? Wait, no, let's re - check. Wait, ∠FGJ and ∠FGE: do they form a linear pair? Wait, maybe I made a mistake. Wait, ∠IJL and ∠IJG: ∠IJL and ∠IJG share the common side IJ and vertex J? Wait, no. Wait, let's re - evaluate each pair:

- ∠KJL and ∠IJL: As IJK is a straight line, JL is a common side, vertex J. So they are adjacent (form a linear pair, which are adjacent and supplementary).
- ∠HGE and ∠FGE: FGH is a straight line, GE is common side, vertex G. So adjacent (linear pair).
- ∠FGJ and ∠FGE: Common vertex G, common side FG. The other sides are GJ and GE. But are they adjacent? Wait, the angle between FG and GJ is ∠FGJ, and between FG and GE is ∠FGE. But GJ and GE are two different lines from G. Wait, maybe not. Wait, ∠IJL and ∠IJG: ∠IJL has sides IJ and JL, ∠IJG has sides IJ and JG. So common side IJ, common vertex J. So they are adjacent. Wait, maybe my initial analysis was wrong. Wait, let's start over.

Adjacent angles: share a common vertex, a common side, and their non - common sides are distinct and lie on either side of the common side.

1. ∠KJL and ∠IJL:
- Common vertex: J
- Common side: JL
- Non - common sides: JK and JI (which are on a straight line, so they are adjacent (linear pair)).
2. ∠HGE and ∠FGE:
- Common vertex: G
- Common side: GE
- Non - common sides: GH and GF (on a straight line, adjacent (linear pair)).
3. ∠FGJ and ∠FGE:
- Common vertex: G
- Common side: FG
- Non - common sides: GJ and GE. Are these two angles adjacent? Let's see, the angle between FG and GJ is ∠FGJ, and between FG and GE is ∠FGE. Since GJ and GE are two different rays from G, and FG is the common side, they are adjacent? Wait, maybe. Wait, no, maybe I messed up. Wait, the line EGL passes through G and J? Wait, the diagram: E - G - J - L? Wait, maybe the line is EGL, with G and J on it. And FGH is a line with F - G - H, and IJK is a line with I - J - K.

So ∠FGJ: vertex G, sides FG and GJ. ∠FGE: vertex G, sides FG and GE. Since GJ and GE are on the same line (EGL), so ∠FGJ and ∠FGE share FG, vertex G, and GJ and GE are on a straight line. So they are adjacent.

Wait, no, ∠IJL and ∠IJG: ∠IJL has sides IJ and JL, ∠IJG has sides IJ and JG. So common side IJ, vertex J. So adjacent.

Wait, maybe the correct pairs are ∠KJL and ∠IJL, ∠HGE and ∠FGE, ∠IJL and ∠IJG, ∠FGJ and ∠FGE? Wait, no, let's check the options given. The options are:

- ∠KJL and ∠IJL
- ∠HGE and ∠FGE
- ∠FGJ and ∠FGE
- ∠IJL and ∠IJG

Wait, let's analyze each option:

- ∠KJL and ∠IJL:
- Common vertex: J
- Common side: JL
- Non - common sides: JK and JI (collinear, so adjacent).
- ∠HGE and ∠FGE:
- Common vertex: G
- Common side: GE
- Non - common sides: GH and GF (collinear, so adjacent).
- ∠FGJ and ∠FGE:
- Common vertex: G
- Common side: FG
- Non - common sides: GJ and GE (collinear, since GJ and GE are on the same line EGL), so adjacent.
- ∠IJL and ∠IJG:
- Common vertex: J
- Common side: IJ
- Non - common sides: JL and JG (not collinear? Wait, JL and JG: JL is part of EGL, JG is part of EGL? Wait, EGL is a straight line, so JG and JL are on the same line. So ∠IJL (sides IJ, JL) and ∠IJG (sides IJ, JG) share IJ, vertex J, and JL and JG are on a straight line. So adjacent.

Wait, but maybe the intended correct ones are ∠KJL and ∠IJL, ∠HGE and ∠FGE, ∠IJL and ∠IJG, ∠FGJ and ∠FGE? But let's check the definition again. Adjacent angles must have a common side, common vertex, and the non - common sides must be on different sides of the common side (i.e., not overlapping).

So:

- ∠KJL and ∠IJL: Yes, adjacent (linear pair).
- ∠HGE and ∠FGE: Yes, adjacent (linear pair).
- ∠FGJ and ∠FGE: Yes, because they share FG, vertex G, and GJ and GE are on a straight line (so non - common sides are on a straight line, so adjacent).
- ∠IJL and ∠IJG: Yes, share IJ, vertex J, and JL and JG are on a straight line (so adjacent).

But maybe the problem has some of these as correct. Wait, the original problem says "Select all the pairs of adjacent angles". Let's go back to the diagram.

Line FGH is horizontal, line EGL is a transversal (or another line) intersecting FGH at G and IJK at J.

∠KJL: between JK and JL.

∠IJL: between IJ and JL. So common side JL, vertex J, non - common sides JK and IJ (which are on a straight line (IJK)), so adjacent.

∠HGE: between GH and GE.

∠FGE: between FG and GE. Common side GE, vertex G, non - common sides GH and FG (on a straight line (FGH)), so adjacent.

∠FGJ: between FG and GJ.

∠FGE: between FG and GE. Common side FG, vertex G, non - common sides GJ and GE (on a straight line (EGL)), so adjacent.

∠IJL: between IJ and JL.

∠IJG: between IJ and JG. Common side IJ, vertex J, non - common sides JL and JG (on a straight line (EGL)), so adjacent.

So all four pairs? But maybe the options are as given, and we need to select the correct ones. Wait, maybe I made a mistake. Let's check the options again:

The options are:

1. ∠KJL and ∠IJL
2. ∠HGE and ∠FGE
3. ∠FGJ and ∠FGE
4. ∠IJL and ∠IJG

Let's confirm each:

1. ∠KJL and ∠IJL: Adjacent (linear pair) - correct.
2. ∠HGE and ∠FGE: Adjacent (linear pair) - correct.
3. ∠FGJ and ∠FGE: Adjacent (share FG, vertex G, GJ and GE are on EGL) - correct.
4. ∠IJL and ∠IJG: Adjacent (share IJ, vertex J, JL and JG are on EGL) - correct.

But maybe the problem has a different intended answer. Wait, maybe the diagram is different. Alternatively, maybe ∠FGJ and ∠FGE are not adjacent. Wait, no, if GJ and GE are on the same line, then ∠FGJ + ∠FGE = ∠JG E? No, ∠FGJ is between FG and GJ, ∠FGE is between FG and GE. If GJ and GE are colinear, then ∠FGJ and ∠FGE are adjacent (form a linear pair? Wait, no, ∠FGJ + ∠FGE would be ∠JG E? No, maybe not. Wait, maybe GJ and GE are not colinear. Wait, maybe the line EGL is E - G - J - L, so GJ is part of EGL. So GE and GJ are on the same line (EGL), so ∠FGJ (FG to GJ) and ∠FGE (FG to GE) share FG, and GJ and GE are on the same line, so they are adjacent (the angle between FG and GJ and between FG and GE, with GJ and GE on a straight line, so they are adjacent and supplementary? Wait, no, adjacent angles just need to share a common side and vertex, and non - common sides on either side of the common side. So yes, they are adjacent.

Similarly, ∠IJL (IJ to JL) and ∠IJG (IJ to JG) share IJ, and JL and JG are on EGL, so adjacent.

So the correct pairs are ∠KJL and ∠IJL, ∠HGE and ∠FGE, ∠FGJ and ∠FGE, ∠IJL and ∠IJG. But maybe the problem has some of them. Wait, maybe the original problem's options are presented as:

- ∠KJL and ∠IJL (correct)
- ∠HGE and ∠FGE (correct)
- ∠FGJ and ∠FGE (correct)
- ∠IJL and ∠IJG (correct)

But maybe I misread the options. Alternatively, maybe the answer is ∠KJL and ∠IJL, ∠HGE and ∠FGE, ∠IJL and ∠IJG, ∠FGJ and ∠FGE.

Answer:

The pairs of adjacent angles are $\angle KJL$ and $\angle IJL$, $\angle HGE$ and $\angle FGE$, $\angle FGJ$ and $\angle FGE$, $\angle IJL$ and $\angle IJG$. (If we need to select from the given options, all four pairs are adjacent. But if there is a mistake in my analysis, please re - check the diagram.)