Question

which angles are consecutive interior angles? ∠dcf and ∠efc, ∠dcf and ∠gfh, ∠dcf and ∠gfc, ∠dcf and ∠bcf

Explanation:

Step1: Recall consecutive interior angles definition

Consecutive interior angles are two angles that lie between two lines (the "interior") and on the same side of the transversal. The two lines are parallel (or not), and the transversal crosses them.

Step2: Analyze each option

• Option 1: $\angle DCF$ and $\angle EFC$: The lines $EB$ and $EG$ (wait, actually $EB$ and $EG$? Wait, the two parallel lines here are $EB$ (or $CD$) and $EF$ (or $EG$)? Wait, the transversal is $HA$ (the line with points $H, F, C, A$). So $\angle DCF$ is between $CD$ and $HA$, and $\angle EFC$ is between $EG$ (or $EF$) and $HA$, on the same side of the transversal. Wait, let's check positions. $CD$ and $EG$ are the two vertical lines (parallel, since they have the same direction). The transversal is $HA$. Consecutive interior angles should be between the two parallel lines and on the same side of the transversal. So $\angle DCF$ is at $C$ between $CD$ and $HA$, $\angle EFC$ is at $F$ between $EG$ and $HA$, and they are on the same side of the transversal (the "inside" of the two parallel lines, and same side of $HA$). Wait, let's check other options:
• $\angle DCF$ and $\angle GFH$: $\angle GFH$ is outside the two parallel lines (since $G$ is below $F$, so $\angle GFH$ is below the transversal, while $\angle DCF$ is above? No, wait, positions. $\angle GFH$ is at $F$, below the transversal, while $\angle DCF$ is at $C$, above? No, maybe not. Wait, $\angle GFH$ is adjacent to $\angle EFC$? Wait, no. $\angle GFH$ is a vertical angle or something else.
• $\angle DCF$ and $\angle GFC$: $\angle GFC$ is a straight angle? No, $\angle GFC$ is at $F$, between $GF$ and $FC$, but $GF$ is part of $EG$, and $FC$ is the transversal. Wait, no, $\angle GFC$ is actually supplementary to $\angle EFC$? Maybe not.
• $\angle DCF$ and $\angle BCF$: These are adjacent angles, forming a linear pair, not consecutive interior angles.

Wait, let's re-express: The two parallel lines are $CD$ (with $B$ and $D$) and $EG$ (with $E$ and $G$). The transversal is $AC$ (the line $H-F-C-A$). So consecutive interior angles must be between $CD$ and $EG$, and on the same side of $AC$. So $\angle DCF$ is at $C$ (between $CD$ and $AC$), and $\angle EFC$ is at $F$ (between $EG$ and $AC$), and they are on the same side of $AC$ (the "inside" of the two parallel lines, and same side of the transversal). So this fits the definition.

Step3: Confirm the correct pair

By the definition of consecutive interior angles (two angles between two lines, on the same side of the transversal), $\angle DCF$ and $\angle EFC$ are between the two parallel lines (CD and EG) and on the same side of the transversal (AC). The other options do not satisfy this: $\angle DCF$ and $\angle GFH$ are not between the lines, $\angle DCF$ and $\angle GFC$ – $\angle GFC$ is not between the lines, $\angle DCF$ and $\angle BCF$ are adjacent (linear pair).

Answer:

$\angle DCF$ and $\angle EFC$