Question

1. given the following two parallel lines cut by a transversal. which pair of angles represents alternate exterior angles? ∠6 and ∠5 ∠7 and ∠5 ∠1 and ∠2 ∠4 and ∠8

Explanation:

Step1: Recall Alternate Exterior Angles Definition

Alternate exterior angles are two angles that lie outside the two parallel lines and on opposite sides of the transversal.

Step2: Analyze Each Option

- **∠6 and ∠5**: Both are inside the parallel lines (interior), so not exterior.
- **∠7 and ∠5**: ∠7 is interior, ∠5 is interior, so no.
- **∠1 and ∠2**: ∠1 is exterior (above top parallel line), ∠2 is interior (below bottom parallel line)? Wait, no, let's check positions. Wait, two parallel horizontal lines, transversal crossing them. ∠1: top right, exterior? Wait, no, alternate exterior: outside the two lines. Let's label: top line (above) and bottom line (below). Transversal crosses them. ∠4: top left (exterior), ∠8: bottom right (exterior)? Wait, no, let's re-express. Wait, the two parallel lines are horizontal. The transversal is a slant line. So exterior angles are those not between the two parallel lines. So ∠1: top right (exterior), ∠2: bottom left (interior? No, ∠2 is below bottom line? Wait, no, the two parallel lines are the two horizontal lines. So between them is the region between the two horizontals. So ∠1: above top horizontal (exterior), ∠4: above top horizontal (left, exterior), ∠3: below bottom horizontal (exterior), ∠8: below bottom horizontal (right, exterior). Wait, maybe I messed up. Wait, alternate exterior angles: lie outside the two parallel lines, on opposite sides of transversal. Let's check each pair:
- ∠4 and ∠8: ∠4 is above top line (left), ∠8 is below bottom line (right). Wait, no, maybe the correct pair is ∠4 and ∠8? Wait, no, the option is ∠4 and ∠8? Wait, the options are ∠6 and ∠5, ∠7 and ∠5, ∠1 and ∠2, ∠4 and ∠8. Wait, let's re-express the angles:

Top horizontal line: intersected by transversal, creating ∠1 (top right), ∠4 (top left), ∠7 (bottom right, between the two lines), ∠6 (bottom left, between the two lines).

Bottom horizontal line: intersected by transversal, creating ∠5 (top right, between the two lines), ∠2 (top left, between the two lines), ∠8 (bottom right, below the line), ∠3 (bottom left, below the line).

So exterior angles: ∠1 (above top line, right), ∠4 (above top line, left), ∠3 (below bottom line, left), ∠8 (below bottom line, right).

Now, alternate exterior: on opposite sides of transversal, outside the two lines.

So ∠4 (above top, left) and ∠8 (below bottom, right): opposite sides of transversal? Wait, transversal is slanting. ∠4 is on left side of transversal (above top line), ∠8 is on right side of transversal (below bottom line). Yes, they are outside the two parallel lines (∠4 is above top, ∠8 is below bottom) and on opposite sides of transversal. Wait, but let's check the options. Wait, the option is ∠4 and ∠8? Wait, the last option is ∠4 and ∠8. Wait, but let's check the other options:

∠1 and ∠2: ∠1 is exterior (above top), ∠2 is interior (between the lines), so no.

∠7 and ∠5: both interior, between the lines.

∠6 and ∠5: both interior, between the lines.

So the only pair where both are exterior and alternate (opposite sides of transversal) is ∠4 and ∠8? Wait, but wait, maybe I made a mistake. Wait, alternate exterior angles: for two parallel lines cut by transversal, alternate exterior angles are congruent, and lie outside the two lines, on opposite sides of transversal. So ∠4 (above top line, left of transversal) and ∠8 (below bottom line, right of transversal) – yes, that's alternate exterior. Wait, but let's check the options again. The options are:

1. ∠6 and ∠5

2. ∠7 and ∠5

3. ∠1 and ∠2

4. ∠4 and ∠8

So the correct pair is ∠4 and ∠8? Wait, but let's confirm. Wait, maybe I mixed up ∠1 and ∠3? No, the options don't have ∠1 and ∠3. Wait, the option is ∠4 and ∠8. So that's the pair of alternate exterior angles.

Answer:

∠4 and ∠8 (the last option, labeled as ∠4 and ∠8)