Question

given: a and b are parallel and c is a transversal. prove: ∠2 ≅ ∠7 use the drop - down menus to complete the paragraph proof showing that alternate interior angles are congruent. we know that lines a and b are parallel and that line c is a transversal because that is given. we can tell that angles 2 and 5 are congruent because vertical angles are congruent. angles 5 and 7 are congruent because drop - down menu angles by parallel lines cut by a drop - down menu ent. therefore, angles 2 and 7 drop - down menu on the transitive property. the drop - down menu options for the second blank include vertical, corresponding, alternate interior, alternate exterior.

Explanation:

Step1: Analyze Angle 5 and 7 Relationship

Lines \(a\) and \(b\) are parallel, cut by transversal \(c\). Angles 5 and 7: when parallel lines are cut by a transversal, alternate interior angles are congruent. Wait, no—wait, angle 5 and angle 7: let's check positions. Line \(b\) and \(a\) are parallel, transversal \(c\). Angle 5 is on line \(b\), angle 7 on line \(a\), both between the two lines (interior) and on opposite sides of transversal (alternate). Wait, but also, corresponding angles? Wait, no—wait, angle 5 and angle 7: actually, angle 5 and angle 7 are alternate interior angles? Wait, no, angle 5 is above line \(b\), angle 7 is below line \(a\)? Wait, no, looking at the diagram: line \(b\) has angles 1,5,2,6; line \(a\) has 3,7,4,8. Transversal \(c\) crosses them. So angle 5 and angle 7: when lines \(a\) and \(b\) are parallel, alternate interior angles would be angle 6 and angle 7? Wait, no, maybe I messed up. Wait, the correct reason for angle 5 and angle 7: actually, angle 5 and angle 7—wait, no, angle 5 and angle 7: let's see, angle 5 is on line \(b\), angle 7 on line \(a\), same position relative to transversal? No, angle 5 is above line \(b\), angle 7 is above line \(a\)? Wait, no, line \(b\) is above line \(a\). So angle 5 and angle 7: are they alternate interior? No, alternate interior would be between the two lines. Wait, angle 2 and angle 7: angle 2 is on line \(b\) (below angle 1), angle 7 on line \(a\) (above angle 4). Wait, the problem is to prove \(\angle 2 \cong \angle 7\). We know \(\angle 2 \cong \angle 5\) (vertical angles). Then \(\angle 5\) and \(\angle 7\): since \(a \parallel b\), alternate interior angles? Wait, no, angle 5 and angle 7: actually, angle 5 and angle 7 are alternate interior angles? Wait, no, angle 5 is above line \(b\), angle 7 is below line \(a\)? Wait, maybe the correct term is alternate interior? Wait, no, the options are vertical, corresponding, alternate interior, alternate exterior. Wait, the drop-down has "alternate interior" as an option (marked with X, but maybe that's the correct one). Wait, no—wait, when two parallel lines are cut by a transversal, alternate interior angles are congruent. So angle 5 and angle 7: are they alternate interior? Let's see: the two parallel lines are \(a\) and \(b\), transversal \(c\). The interior is between \(a\) and \(b\). So angle 2 (on \(b\), between \(b\) and \(a\)) and angle 7 (on \(a\), between \(b\) and \(a\))—wait, no, angle 2 is on line \(b\) (below angle 1), angle 7 is on line \(a\) (above angle 4). So the region between \(a\) and \(b\) is the interior. So angle 2 (on \(b\), interior) and angle 7 (on \(a\), interior), on opposite sides of transversal \(c\): that's alternate interior. But angle 5 is on line \(b\), above the interior (exterior?), no—wait, line \(b\) has angles 1 (top left), 5 (top right), 2 (bottom left), 6 (bottom right). Line \(a\) has 3 (top left), 7 (top right), 4 (bottom left), 8 (bottom right). So transversal \(c\) is the line with the arrows. So angle 5 (top right of \(b\)) and angle 7 (top right of \(a\)): those are corresponding angles! Oh! Because they are in the same position relative to the parallel lines and transversal. So angle 5 (on \(b\), top right) and angle 7 (on \(a\), top right) are corresponding angles. So when \(a \parallel b\), corresponding angles are congruent. So that's the reason: corresponding angles. Wait, but the option has "corresponding" as an option. So the drop-down for "Angles 5 and 7 are congruent because [ ] angles by parallel lines cut by a transversal are congruent." So the correct term is "alternate interior"? Wait, no, let's re-express:

Wait, the diagram:

- Line \(b\) (top) and line \(a\) (bottom) are parallel.
- Transversal \(c\) crosses them.
- Angle 5: top right of line \(b\), between transversal and line \(b\).
- Angle 7: top right of line \(a\), between transversal and line \(a\).

So these are corresponding angles (same position relative to parallel lines and transversal), so corresponding angles are congruent when lines are parallel.

Wait, but earlier thought: angle 2 and angle 5 are vertical angles (correct, since they are opposite each other at the intersection of line \(b\) and transversal \(c\)). Then angle 5 and angle 7: since \(a \parallel b\), corresponding angles are congruent. Then by transitive property, angle 2 (congruent to 5) and angle 7 (congruent to 5) are congruent. Wait, but the drop-down options include "alternate interior", "corresponding", etc.

Wait, the problem's drop-down for the second blank: the options are vertical, corresponding, alternate interior, alternate exterior.

We need to find why \(\angle 5 \cong \angle 7\). Since \(a \parallel b\), and transversal \(c\), \(\angle 5\) and \(\angle 7\) are corresponding angles (same position: top right of each parallel line, relative to transversal). So the correct reason is "corresponding" angles are congruent. Wait, but wait, angle 5 and angle 7: are they alternate interior? No, alternate interior would be between the two lines. Angle 5 is above line \(b\) (exterior to the space between \(a\) and \(b\)), angle 7 is above line \(a\) (exterior? No, line \(b\) is above line \(a\), so the space between \(b\) and \(a\) is the interior. So angle 5 is above \(b\) (exterior), angle 7 is above \(a\) (exterior). Wait, no, angle 2 is below \(b\) (interior, between \(b\) and \(a\)), angle 7 is above \(a\) (interior? No, line \(a\) is the bottom parallel line, so above \(a\) is between \(a\) and \(b\) (interior), and below \(a\) is exterior. So angle 7 is above \(a\) (interior), angle 5 is below \(b\)? Wait, no, line \(b\) is the top line, so below \(b\) is between \(b\) and \(a\) (interior), above \(b\) is exterior. So angle 5 is above \(b\) (exterior), angle 2 is below \(b\) (interior), angle 6 is below \(b\) (interior, opposite angle 2), angle 1 is above \(b\) (exterior, opposite angle 5). Then line \(a\): angle 7 is above \(a\) (interior, between \(a\) and \(b\)), angle 3 is below \(a\) (exterior), angle 8 is below \(a\) (exterior, opposite angle 3), angle 4 is below \(a\) (exterior? Wait, no, maybe I got the interior/exterior wrong.

Alternative approach: the theorem is that if two parallel lines are cut by a transversal, then alternate interior angles are congruent, corresponding angles are congruent, alternate exterior angles are congruent.

Corresponding angles: same position relative to the parallel lines and transversal (e.g., top right, top left, bottom right, bottom left of each parallel line).

Alternate interior: between the two lines, opposite sides of transversal.

Alternate exterior: outside the two lines, opposite sides of transversal.

So angle 5: position: top right of line \(b\) (relative to transversal \(c\)).

Angle 7: position: top right of line \(a\) (relative to transversal \(c\)).

So same position: corresponding angles. Therefore, \(\angle 5 \cong \angle 7\) because corresponding angles are congruent when lines are parallel.

So the correct option for the drop-down is "corresponding".

Wait, but the drop-down in the image has "alternate interior" marked with X, but maybe that's a mistake. Wait, no, let's check again. The problem is to prove \(\angle 2 \cong \angle 7\). We have \(\angle 2 \cong \angle 5\) (vertical angles). Then \(\angle 5\) and \(\angle 7\): if \(a \parallel b\), then alternate interior angles? Wait, angle 2 is on line \(b\) (between \(b\) and \(a\), left of transversal), angle 7 is on line \(a\) (between \(b\) and \(a\), right of transversal). So angle 2 and angle 7 are alternate interior angles? Wait, no, angle 2 and angle 7: angle 2 is on line \(b\), angle 7 on line \(a\), both between the two lines (interior) and on opposite sides of transversal (alternate). So angle 2 and angle 7 are alternate interior angles. But the proof is using \(\angle 2 \cong \angle 5\) (vertical) and \(\angle 5 \cong \angle 7\) (then transitive). Wait, \(\angle 5\) and \(\angle 7\): where is angle 5? Angle 5 is on line \(b\), above the interior (exterior?), no—wait, line \(b\) is above line \(a\), so the interior is between \(b\) and \(a\). So angle 5 is above line \(b\) (exterior), angle 7 is above line \(a\) (interior). Wait, no, maybe the diagram is different. Let's look at the diagram:

- Line \(b\) (top) has angles 1 (top left), 5 (top right), 2 (bottom left), 6 (bottom right).

- Line \(a\) (bottom) has angles 3 (top left), 7 (top right), 4 (bottom left), 8 (bottom right).

- Transversal \(c\) crosses them, with arrows going up and down.

So angle 2 is bottom left of line \(b\) (between transversal and line \(b\), below angle 1), angle 7 is top right of line \(a\) (between transversal and line \(a\), above angle 4).

So angle 2 and angle 7: angle 2 is on line \(b\) (upper parallel), angle 7 on line \(a\) (lower parallel), both between the two parallels (interior) and on opposite sides of transversal (alternate). So they are alternate interior angles. But the proof is using \(\angle 2 \cong \angle 5\) (vertical angles: angle 2 and angle 5 are vertical, since they are opposite at the intersection of line \(b\) and transversal \(c\)). Then \(\angle 5\) and \(\angle 7\): angle 5 is top right of line \(b\), angle 7 is top right of line \(a\) – same position, so corresponding angles. Therefore, \(\angle 5 \cong \angle 7\) (corresponding angles, since \(a \parallel b\)). Then by transitive property, \(\angle 2 \cong \angle 7\).

So the correct reason for \(\angle 5 \cong \angle 7\) is "corresponding" angles.

Therefore, the drop-down should be "corresponding".

Step2: Confirm the Reason

The key is that when two parallel lines are cut by a transversal, corresponding angles (same position relative to lines and transversal) are congruent. Angle 5 and angle 7 are in the same position (top right) relative to their respective parallel lines (\(b\) and \(a\)) and transversal \(c\), so they are corresponding angles. Thus, the correct option for the drop-down is "corresponding".

Answer:

corresponding