two parallel lines are intersected by a third line so that angles 1 and 5 are congruent.
which statement is true about angles 3 and 5?
they are acute
they are congruent
they are complementary
they are supplementary

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# Brief Explanations:
1. First, recall the properties of parallel lines cut by a transversal. When two parallel lines are cut by a transversal, consecutive interior angles are supplementary, corresponding angles are congruent, etc. Also, vertical angles are congruent, and linear pairs are supplementary.
 - Angle 1 and angle 3 are vertical angles, so $\angle1=\angle3$.
 - Given that $\angle1\cong\angle5$, by the transitive property (since $\angle1 = \angle3$ and $\angle1=\angle5$), we get $\angle3=\angle5$. But we also need to check the relationship between $\angle3$ and $\angle5$ in terms of their sum or congruence. Wait, actually, $\angle3$ and $\angle5$: let's look at the linear pair and consecutive interior angles. Wait, $\angle3$ and $\angle4$ are a linear pair (supplementary), $\angle5$ and $\angle6$ are a linear pair. But also, $\angle3$ and $\angle5$: since the lines are parallel, and the transversal, $\angle3$ and $\angle5$ are same - side interior angles? Wait no, wait the diagram: angle 3 and angle 5, let's see the positions. Wait, angle 3 and angle 5: angle 3 is above the lower parallel line and below the upper parallel line, angle 5 is below the upper parallel line (wait no, the upper parallel line has angles 1,2,3,4; lower has 5,6. The transversal crosses them. So angle 3 and angle 5: angle 3 and angle 5 are same - side interior angles? Wait no, actually, angle 3 and angle 5: let's check the sum. Since angle 3 and angle 4 are supplementary ($\angle3+\angle4 = 180^{\circ}$), and angle 4 and angle 5: if the lines are parallel, angle 4 and angle 5 are corresponding angles? Wait no, angle 1 and angle 5 are congruent (given), angle 1 and angle 3 are vertical angles (so $\angle1=\angle3$), so $\angle3=\angle5$. But also, angle 3 and angle 5: let's see, angle 3 and angle 5 are same - side interior angles? Wait, no, same - side interior angles are supplementary. Wait, I think I made a mistake. Wait, the two parallel lines: upper line and lower line. The transversal cuts them. Angle 3 is on the upper line, below the transversal; angle 5 is on the lower line, below the transversal (wait no, the lower line's angle 5 is on the left - hand side of the transversal). Wait, actually, angle 3 and angle 5: let's use the linear pair and the given. Wait, angle 1 and angle 5 are congruent (given). Angle 1 and angle 3 are vertical angles (so $\angle1=\angle3$). So $\angle3=\angle5$. But also, angle 3 and angle 5: let's check their sum. Wait, angle 3 and angle 5: if we look at the linear pair, angle 3 and angle 4 are supplementary ($\angle3+\angle4 = 180^{\circ}$), and angle 4 and angle 5: if the lines are parallel, angle 4 and angle 5 are corresponding angles? Wait no, angle 4 and angle 6 are corresponding angles. Wait, maybe a better approach: angle 3 and angle 5. Since angle 3 and angle 4 are supplementary ($\angle3+\angle4 = 180^{\circ}$), and angle 4 and angle 5: if the lines are parallel, angle 4 and angle 5 are same - side interior angles? No, same - side interior angles are supplementary. Wait, I think I messed up. Let's start over.

Given that two parallel lines are cut by a transversal. Angles 1 and 5 are congruent (given). Angle 1 and angle 3 are vertical angles, so $\angle1=\angle3$. Therefore, $\angle3=\angle5$ (by substitution). Now, angle 3 and angle 5: let's check the relationship. Wait, angle 3 and angle 5: are they supplementary? Wait, no, if $\angle3=\angle5$, but also, angle 3 and angle 4 are supplementary ($\angle3+\angle4 = 180^{\circ}$), and angle 4 and angle 5: if the lines are parallel, angle 4 and angle 5 are same - side interior angles, so they should be supplementary. Wait, that means $\angle4+\angle5 = 180^{\circ}$, and since $\angle3+\angle4 = 180^{\circ}$, then $\angle3=\angle5$. But also, angle 3 and angle 5: let's see, angle 3 and angle 5 are same - side interior angles? Wait, no, same - side interior angles are supplementary. Wait, I think I confused the angles. Let's look at the diagram again. The upper parallel line has angles 1 (top - left), 2 (top - right), 3 (bottom - left), 4 (bottom - right). The lower parallel line has angles 5 (top - left), 6 (top - right). The transversal goes from bottom - left to top - right. So angle 3 is adjacent to angle 4 (linear pair), angle 5 is adjacent to angle 6 (linear pair). Now, angle 3 and angle 5: angle 3 is below the upper parallel line and to the left of the transversal; angle 5 is above the lower parallel line and to the left of the transversal. So angle 3 and angle 5 are same - side interior angles. Wait, same - side interior angles are supplementary. Wait, that's the key. So $\angle3+\angle5 = 180^{\circ}$. So they are supplementary.

Wait, let's correct the earlier mistake. Given that angle 1 and angle 5 are congruent (given). Angle 1 and angle 3 are vertical angles ($\angle1=\angle3$). So $\angle3=\angle5$ (by substitution). But also, angle 3 and angle 4 are supplementary ($\angle3+\angle4 = 180^{\circ}$), and angle 4 and angle 5: since the lines are parallel, angle 4 and angle 5 are same - side interior angles, so $\angle4+\angle5 = 180^{\circ}$. But we also know that $\angle3=\angle5$, so substituting $\angle3$ for $\angle5$ in $\angle4+\angle5 = 180^{\circ}$, we get $\angle4+\angle3 = 180^{\circ}$, which is consistent with them being a linear pair. But the relationship between $\angle3$ and $\angle5$: since they are same - side interior angles (between two parallel lines cut by a transversal), they are supplementary. So $\angle3+\angle5 = 180^{\circ}$, which means they are supplementary.

2. Now let's analyze each option:
 - Option 1: They are acute. We don't know if they are acute. They could be right angles or obtuse angles (if the transversal is perpendicular, they would be right angles; if the transversal is not, one could be acute and one obtuse, but since they are supplementary, if one is acute, the other is obtuse. So this is false.
 - Option 2: They are congruent. We thought earlier $\angle3=\angle5$, but that was a mistake. Because $\angle3$ and $\angle5$ are same - side interior angles, so they are supplementary, not congruent (unless they are right angles, but we can't assume that). So this is false.
 - Option 3: They are complementary. Complementary angles sum to $90^{\circ}$, but we know they sum to $180^{\circ}$, so this is false.
 - Option 4: They are supplementary. Since they are same - side interior angles between two parallel lines cut by a transversal, their sum is $180^{\circ}$, so they are supplementary. This is true.

# Answer:
They are supplementary.

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