Question

multiple choice 9 points
in the figure shown at the right, lines l and m are parallel.
what type of angles do the angle pair of ∠2 and ∠4 form?
○ alternate exterior
○ same - side interior
○ alternate interior
○ linear pair
○ vertical angles
○ corresponding angles
○ same - side exterior

Explanation:

Step1: Recall Angle Pair Definitions

- Alternate exterior: Angles on opposite sides of transversal, outside parallel lines.
- Same - side interior: Angles on same side of transversal, inside parallel lines.
- Alternate interior: Angles on opposite sides of transversal, inside parallel lines.
- Linear pair: Adjacent angles forming a straight line (sum to \(180^\circ\)).
- Vertical angles: Opposite angles formed by intersecting lines.
- Corresponding angles: Angles in same relative position at each intersection.
- Same - side exterior: Angles on same side of transversal, outside parallel lines.

Step2: Analyze \(\angle2\) and \(\angle4\)

- \(\angle2\) and \(\angle4\): \(\angle2\) is inside lines \(l\) and \(m\) (between them), \(\angle4\) is also inside. They are on the same side of the transversal (the vertical line intersecting \(l\) and \(m\)) and adjacent, forming a straight line (since \(l\) and \(m\) are parallel and the transversal intersects them, \(\angle2+\angle3 = 180^\circ\) and \(\angle3+\angle4=180^\circ\), but more directly, \(\angle2\) and \(\angle4\) are adjacent and form a linear pair? Wait, no, wait. Wait, \(\angle2\) and \(\angle3\) are same - side interior? No, wait, let's re - examine. Wait, the transversal is the vertical line. \(\angle2\) is at the intersection of \(l\) and transversal, below \(l\), left of transversal. \(\angle4\) is at intersection of \(m\) and transversal, below \(m\), left of transversal? No, wait, no. Wait, \(\angle2\) and \(\angle4\): \(\angle2\) is between \(l\) and \(m\), left of transversal. \(\angle4\) is below \(m\), left of transversal? No, no, the diagram: line \(l\) is top parallel, \(m\) is bottom parallel. Transversal is vertical. \(\angle2\) is between \(l\) and \(m\), left of transversal. \(\angle3\) is between \(l\) and \(m\), left of transversal (wait, no, \(\angle3\) is between \(l\) and \(m\), right? Wait, no, labels: \(l\) has \(\angle1\) (top left), \(\angle2\) (bottom left); \(m\) has \(\angle3\) (bottom left), \(\angle4\) (bottom right? Wait, no, the diagram: \(l\) is horizontal, left arrow on left, right arrow on right. Transversal is vertical, up arrow top, down arrow bottom. At \(l\) - transversal intersection: \(\angle1\) (top left), \(\angle5\) (top right), \(\angle2\) (bottom left), \(\angle6\) (bottom right). At \(m\) - transversal intersection: \(\angle3\) (bottom left), \(\angle7\) (bottom right), \(\angle4\) (wait, no, maybe I mislabeled. Wait, standard labeling: when two lines are intersected by a transversal, the angles are labeled in order. So at \(l\): top left \(\angle1\), top right \(\angle5\), bottom left \(\angle2\), bottom right \(\angle6\). At \(m\): top left \(\angle3\), top right \(\angle7\), bottom left \(\angle4\)? No, bottom right \(\angle8\). Wait, no, the correct way: \(\angle3\) is below \(m\) left, \(\angle4\) is below \(m\) right? No, the user's diagram: \(l\) has \(\angle1\) (top left), \(\angle5\) (top right), \(\angle2\) (bottom left), \(\angle6\) (bottom right). \(m\) has \(\angle3\) (bottom left), \(\angle7\) (bottom right), \(\angle4\) (wait, no, the labels are \(3\) and \(7\) above \(m\) (between \(l\) and \(m\)) and \(4\) and \(8\) below \(m\). So \(\angle2\) is between \(l\) and \(m\), left of transversal. \(\angle4\) is below \(m\), left of transversal? No, \(\angle3\) is between \(l\) and \(m\), left of transversal; \(\angle4\) is below \(m\), left of transversal? No, I think I made a mistake. Wait, a linear pair is two adjacent angles that form a straight line (their non - common sides are opposite rays). \(\angle2\) and \(\angle3\) are same - side interior? No, \(\angle2\) and \(\angle4\): \(\angle2\) and \(\angle4\) are adjacent, with a common side (the left side of the transversal) and their non - common sides form a straight line (since \(l\) and \(m\) are parallel, the line containing \(l\) and \(m\) are horizontal, and the transversal is vertical. So \(\angle2\) is above \(\angle4\), and they form a linear pair (sum to \(180^\circ\)). Wait, but the options: linear pair is an option. Wait, but let's check again. Wait, the original options: the selected option was "Corresponding angles", but that's wrong. Wait, no, wait, maybe I misanalyzed. Wait, \(\angle2\) and \(\angle4\): \(\angle2\) is at \(l\) - transversal, \(\angle4\) is at \(m\) - transversal. Wait, no, \(\angle2\) and \(\angle4\) are vertical angles? No, vertical angles are opposite. Wait, no, \(\angle2\) and \(\angle6\) are vertical? No, \(\angle1\) and \(\angle2\) are linear pair, \(\angle3\) and \(\angle4\) are linear pair. \(\angle2\) and \(\angle4\): let's see the positions. \(\angle2\) is between \(l\) and \(m\), left of transversal. \(\angle4\) is below \(m\), left of transversal? No, \(m\) is the lower parallel line. So between \(l\) and \(m\) is the interior. \(\angle2\) is interior (between \(l\) and \(m\)), \(\angle4\) is exterior (below \(m\))? No, I think I messed up the diagram labels. Wait, the correct way: when two parallel lines \(l\) (top) and \(m\) (bottom) are cut by a transversal (vertical line), the angles:

- At \(l\) - transversal intersection:
- Top left: \(\angle1\)
- Top right: \(\angle5\)
- Bottom left: \(\angle2\)
- Bottom right: \(\angle6\)
- At \(m\) - transversal intersection:
- Top left: \(\angle3\)
- Top right: \(\angle7\)
- Bottom left: \(\angle4\)
- Bottom right: \(\angle8\)

Now, \(\angle2\) and \(\angle4\): \(\angle2\) is bottom left at \(l\) - transversal, \(\angle4\) is bottom left at \(m\) - transversal? No, \(\angle4\) is bottom left at \(m\) - transversal? No, \(\angle3\) is top left at \(m\) - transversal, \(\angle4\) is bottom left at \(m\) - transversal. So \(\angle2\) (bottom left at \(l\)) and \(\angle4\) (bottom left at \(m\)): they are in the same relative position (bottom left) at each intersection, so they are corresponding angles? Wait, corresponding angles are in the same relative position. So \(\angle1\) and \(\angle3\) are corresponding, \(\angle2\) and \(\angle4\) are corresponding, \(\angle5\) and \(\angle7\) are corresponding, \(\angle6\) and \(\angle8\) are corresponding. Ah, that's it! So \(\angle2\) and \(\angle4\) are in the same relative position (bottom left) at the intersection of \(l\) - transversal and \(m\) - transversal, so they are corresponding angles.

Answer:

The correct answer is the option labeled "Corresponding angles" (the one with the filled circle). But wait, earlier analysis about linear pair was wrong. The key is that corresponding angles are in the same relative position at each intersection of the transversal with the parallel lines. So \(\angle2\) (at \(l\) - transversal, bottom left) and \(\angle4\) (at \(m\) - transversal, bottom left) are corresponding angles.