Question

df and gi are parallel lines.
which angles are alternate exterior angles?
∠dec and ∠ghj ∠ihe and ∠ghe
∠fec and ∠feh ∠ihj and ∠dec

Explanation:

Step1: Recall alternate exterior angles definition

Alternate exterior angles are formed when a transversal crosses two parallel lines. They lie outside the two parallel lines and on opposite sides of the transversal.

Step2: Analyze each option

- Option 1: $\angle DEC$ and $\angle GHJ$: $\angle DEC$ is above line $DF$ and left of transversal $CJ$, $\angle GHJ$ is below line $GI$ and left of transversal $CJ$. Wait, no, let's re - check. Wait, $DF\parallel GI$, transversal is $CJ$. $\angle DEC$: outside $DF$ and $GI$? Wait, $DF$ and $GI$ are parallel. $\angle DEC$: line $DF$ is upper parallel, $\angle GHJ$: line $GI$ is lower parallel. $\angle DEC$ is on the outer side of $DF$ (left side of transversal, above $DF$), $\angle GHJ$ is on the outer side of $GI$ (left side of transversal, below $GI$)? No, maybe I made a mistake. Wait, let's check the other options.
- Option 2: $\angle IHE$ and $\angle GHE$: These are adjacent angles, not alternate exterior.
- Option 3: $\angle FEC$ and $\angle FEH$: These are adjacent angles, not alternate exterior.
- Option 4: $\angle IHJ$ and $\angle DEC$: $\angle IHJ$ is outside line $GI$ (right side of transversal, below $GI$) and $\angle DEC$ is outside line $DF$ (left side of transversal, above $DF$)? Wait, no. Wait, correct analysis: For two parallel lines $DF$ and $GI$ cut by transversal $CJ$. Alternate exterior angles should be on opposite sides of the transversal and outside the two parallel lines. $\angle DEC$: above $DF$, left of transversal. $\angle IHJ$: below $GI$, right of transversal? No, wait, maybe the first option. Wait, let's re - define: Alternate exterior angles are two angles that lie outside the two parallel lines, on opposite sides of the transversal. So for $DF\parallel GI$, transversal $CJ$. $\angle DEC$: outside $DF$ (since $DF$ is between the two intersection points $E$ and $H$), and $\angle GHJ$: outside $GI$ (since $GI$ is between $E$ and $H$). And they are on opposite sides of the transversal? Wait, no. Wait, maybe the correct pair is $\angle DEC$ and $\angle GHJ$. Wait, let's check the positions again. Line $DF$ (points $D, E, F$) and line $GI$ (points $G, H, I$) are parallel. Transversal is $CJ$ (points $C, E, H, J$). $\angle DEC$: at $E$, between $D - E - F$ and $C - E - J$. $\angle GHJ$: at $H$, between $G - H - I$ and $C - H - J$. $\angle DEC$ is above $DF$ (outside the region between $DF$ and $GI$) and $\angle GHJ$ is below $GI$ (outside the region between $DF$ and $GI$), and they are on opposite sides of the transversal? Wait, no, maybe I messed up. Wait, the other options: $\angle IHJ$ and $\angle DEC$: $\angle IHJ$ is at $H$, below $GI$, right of transversal. $\angle DEC$ is at $E$, above $DF$, left of transversal. These are outside the two parallel lines and on opposite sides of the transversal. Wait, maybe I made a mistake in the first analysis. Let's re - check the definition. Alternate exterior angles: non - adjacent, outside the two parallel lines, on opposite sides of the transversal. So for $DF\parallel GI$, transversal $CJ$. Angles outside $DF$ and $GI$: $\angle DEC$ (outside $DF$, left of transversal), $\angle IHJ$ (outside $GI$, right of transversal) – no, that's same side? Wait, no. Wait, $\angle DEC$: when we look at transversal $CJ$, $\angle DEC$ is on the left side of $CJ$, above $DF$. $\angle GHJ$: on the left side of $CJ$, below $GI$. No, that's same side. Wait, maybe the correct answer is $\angle DEC$ and $\angle GHJ$? Wait, no, let's check the options again. Wait, the first option is $\angle DEC$ and $\angle GHJ$. Let's see: $DF\parallel GI$, transversal $CJ$. $\angle DEC$: exterior to $DF$ (above $DF$), $\angle GHJ$: exterior to $GI$ (below $GI$), and they are on opposite sides of the transversal? Wait, no, they are on the same side (left side of transversal). Wait, I think I made a mistake. Let's check the fourth option: $\angle IHJ$ and $\angle DEC$. $\angle IHJ$: at $H$, below $GI$, right of transversal. $\angle DEC$: at $E$, above $DF$, left of transversal. These are outside the two parallel lines ($DF$ and $GI$) and on opposite sides of the transversal ($CJ$). So they are alternate exterior angles. Wait, but let's confirm with the definition. Alternate exterior angles are equal when lines are parallel, and they lie outside the two parallel lines and on opposite sides of the transversal. So $\angle DEC$ is outside $DF$ (above $DF$) and $\angle IHJ$ is outside $GI$ (below $GI$), and they are on opposite sides of the transversal $CJ$ (left and right). So this pair satisfies the definition. Wait, but maybe the first option is correct. Wait, I think I confused the sides. Let's take a step back. The two parallel lines are $DF$ (top) and $GI$ (bottom). The transversal is $CJ$ (going from $C$ (top) to $J$ (bottom)). Exterior angles: angles not between $DF$ and $GI$. So above $DF$ or below $GI$. Alternate: on opposite sides of the transversal. So for angle above $DF$ (left of transversal: $\angle DEC$), the alternate exterior angle should be below $GI$ (right of transversal: $\angle IHJ$). Yes, so $\angle DEC$ and $\angle IHJ$? Wait, the fourth option is $\angle IHJ$ and $\angle DEC$. So that's the correct pair.

Answer:

$\angle IHJ$ and $\angle DEC$ (the fourth option: $\boldsymbol{\angle IHJ}$ and $\boldsymbol{\angle DEC}$)