two parallel lines are intersected by a third line so that angles 1 and 5 are congruent.\nwhich statement is…

Explanation:

Step1: Identify Angle Relationships

Angles 1 and 3 are vertical angles, so ( \angle 1 \cong \angle 3 ). Given ( \angle 1 \cong \angle 5 ), by transitivity, ( \angle 3 \cong \angle 5 ). Also, angles 3 and 5 are same - side interior angles? Wait, no, let's re - check. Wait, the two parallel lines are cut by a transversal. Angles 3 and 5: let's see the positions. Wait, angle 1 and angle 5 are congruent, angle 1 and angle 3 are vertical angles (so ( \angle 1=\angle 3 )). So ( \angle 3 = \angle 5 ). But also, angles 3 and 5: when two parallel lines are cut by a transversal, consecutive interior angles (same - side interior angles) are supplementary? Wait, no, wait the diagram: the upper line has angles 1,2,3,4; the lower line has 5,6. The transversal crosses both. Angle 3 and angle 5: let's see, angle 3 and angle 4 are supplementary (linear pair), angle 4 and angle 6 are corresponding angles (since lines are parallel), angle 6 and angle 5 are supplementary (linear pair). Wait, maybe a better approach: angle 3 and angle 5: since ( \angle 1\cong\angle 5 ) and ( \angle 1\cong\angle 3 ) (vertical angles), so ( \angle 3\cong\angle 5 ), but also, angle 3 and angle 5 are same - side interior angles? Wait, no, same - side interior angles add up to 180 degrees. Wait, maybe I made a mistake. Wait, the two parallel lines: the upper line and the lower line. The transversal: the line with the arrow. So angle 3 is on the upper line, below the transversal, and angle 5 is on the lower line, above the transversal? Wait, no, looking at the diagram: angle 3 is adjacent to angle 1 (vertical angle), angle 5 is on the lower parallel line. Wait, actually, angle 3 and angle 5: when two parallel lines are cut by a transversal, consecutive interior angles (same - side interior angles) are supplementary. Let's confirm: angle 3 and angle 5 are same - side interior angles. Because they are on the same side of the transversal and between the two parallel lines. So by the consecutive interior angles theorem, if two parallel lines are cut by a transversal, then consecutive interior angles are supplementary. So ( \angle 3+\angle 5 = 180^{\circ} ). Wait, but earlier we thought ( \angle 3\cong\angle 5 ) because ( \angle 1\cong\angle 5 ) and ( \angle 1\cong\angle 3 ). There is a contradiction here, which means I misidentified the angles. Wait, let's label the diagram properly. The upper horizontal line: left - to - right, angles at the intersection with the transversal: angle 1 (top left), angle 2 (top right), angle 3 (bottom left), angle 4 (bottom right). The lower horizontal line: angle 5 (top left), angle 6 (top right). The transversal is the line with the two arrows (one going up - right, one going down - left). So angle 1 and angle 5: are they corresponding angles? If angle 1 is top - left on the upper line, angle 5 is top - left on the lower line, so they are corresponding angles. So if corresponding angles are congruent, the lines are parallel (which they are). Now, angle 3 and angle 5: angle 3 is bottom - left on the upper line, angle 5 is top - left on the lower line. So angle 3 and angle 5: let's see, angle 3 and angle 4 are supplementary (linear pair, ( \angle 3+\angle 4 = 180^{\circ} )). Angle 4 and angle 6 are corresponding angles (so ( \angle 4=\angle 6 )). Angle 6 and angle 5 are supplementary (linear pair, ( \angle 5+\angle 6 = 180^{\circ} )). So from ( \angle 3+\angle 4 = 180^{\circ} ) and ( \angle 5+\angle 6 = 180^{\circ} ) and ( \angle 4=\angle 6 ), we can substitute ( \angle 4 ) with ( \angle 6 ) in the first equation: ( \angle 3+\angle 6 = 180^{\circ} ), and since ( \angle 5+\angle 6 = 180^{\circ} ), then ( \angle 3=\angle 5 )? No, that's not right. Wait, no, ( \angle 3+\angle 4 = 180 ), ( \angle 4=\angle 6 ) (corresponding), ( \angle 5+\angle 6 = 180 ). So ( \angle 3+\angle 4=\angle 5+\angle 6 ), and since ( \angle 4 = \angle 6 ), then ( \angle 3=\angle 5 ). But also, angle 3 and angle 5 are same - side interior angles? Wait, no, same - side interior angles are on the same side of the transversal and between the two lines. Angle 3 is between the two lines, on the left side of the transversal. Angle 5 is also between the two lines, on the left side of the transversal. So they are same - side interior angles. But same - side interior angles of parallel lines are supplementary. So there is a conflict here? Wait, no, I must have mislabeled the angles. Wait, maybe angle 3 and angle 5 are not same - side interior angles. Let's re - draw mentally: upper line: horizontal, left - right. Transversal: a line going from bottom - left to top - right. So angle 1: top - left of upper line, angle 2: top - right, angle 3: bottom - left, angle 4: bottom - right. Lower line: horizontal, left - right. Angle 5: top - left of lower line, angle 6: top - right. So the transversal is going from bottom - left (below lower line) to top - right (above upper line). So angle 3 is on the upper line, below the transversal, to the left. Angle 5 is on the lower line, above the transversal, to the left. So the two angles (3 and 5) are between the two parallel lines and on the same side of the transversal (left side). So they are same - side interior angles. For parallel lines, same - side interior angles are supplementary. So ( \angle 3+\angle 5 = 180^{\circ} ), which means they are supplementary.

Step2: Evaluate the Options

• Option 1: They are acute. We don't know if they are acute. They could be right angles or obtuse, so this is false.
• Option 2: They are congruent. From the earlier wrong approach, but actually, since they are same - side interior angles of parallel lines, they are supplementary, not congruent (unless they are right angles, but we don't know that). So this is false.
• Option 3: They are complementary. Complementary angles add up to ( 90^{\circ} ), but we know they add up to ( 180^{\circ} ) (if they are same - side interior angles of parallel lines), so this is false.
• Option 4: They are supplementary. Since they are same - side interior angles of two parallel lines cut by a transversal, by the consecutive interior angles theorem, they are supplementary (( \angle 3+\angle 5 = 180^{\circ} )). Also, from the given ( \angle 1\cong\angle 5 ) and ( \angle 1 ) and ( \angle 3 ) are supplementary (linear pair: ( \angle 1+\angle 3 = 180^{\circ} )), so substituting ( \angle 5 ) for ( \angle 1 ), we get ( \angle 5+\angle 3 = 180^{\circ} ), so they are supplementary.

Answer:

They are supplementary. (The option "They are supplementary" is the correct one. If we consider the options as:

• They are acute.
• They are congruent.
• They are complementary.
• They are supplementary.

The correct option is the one stating "They are supplementary".)