Question

select all angles that form a pair of alternate exterior angles with ∠5.
∠1
∠4
∠6
∠7

Explanation:

Step1: Recall Alternate Exterior Angles

Alternate exterior angles are formed when a transversal crosses two parallel lines (or in this case, two sets of perpendicular lines, which are parallel to each other in their respective directions). They lie outside the two lines and on opposite sides of the transversal.

Step2: Analyze ∠5's Position

∠5 is in the upper intersection, in the upper - right quadrant (exterior relative to the two lines formed by the intersections). We need to find an angle in the lower intersection that is outside the two lines (formed by the horizontal and vertical lines at the lower intersection) and on the opposite side of the transversal (the vertical line connecting the two intersections).

Step3: Evaluate Each Angle

- ∠1: It is in the lower - right quadrant, inside the "corner" of the lower intersection, not exterior in the correct sense for alternate exterior with ∠5.
- ∠4: It is in the lower - right, adjacent to ∠1, not alternate exterior.
- ∠6: It is in the lower - left quadrant, but let's check the transversal. The vertical line is the transversal. ∠5 is upper - right, ∠7 is lower - left, outside the two horizontal lines (the horizontal lines at each intersection) and on opposite sides of the vertical transversal? Wait, no, let's re - examine. Wait, the two vertical lines (the ones with the arrows) are parallel? Wait, no, the horizontal lines (the ones with the left - right arrows) are parallel, and the vertical lines (up - down arrows) are parallel. Wait, ∠5 is at the upper intersection, between the upper vertical and upper horizontal (right side). The transversal here is the vertical line (connecting the two intersections). The alternate exterior angle should be outside the two horizontal lines (the upper horizontal and lower horizontal) and on the opposite side of the transversal. ∠7 is at the lower intersection, between the lower vertical and lower horizontal (left side), outside the two horizontal lines and on the opposite side of the vertical transversal from ∠5. Wait, maybe I made a mistake earlier. Wait, let's re - define: the two horizontal lines (the left - right arrows) are parallel, and the vertical line (up - down arrows) is the transversal. ∠5 is above the upper horizontal line? No, ∠5 is at the upper intersection, in the upper - right of the upper intersection (between upper vertical (up arrow) and upper horizontal (right arrow)). The lower intersection has horizontal (left - right) and vertical (down arrow) lines. The transversal is the vertical line (connecting the two intersections). So, ∠5 is on the right side of the transversal (vertical line) and above the upper horizontal line. An alternate exterior angle should be on the left side of the transversal and below the lower horizontal line? No, maybe the two vertical lines are parallel? Wait, no, the horizontal lines (left - right) are parallel, and the vertical lines (up - down) are parallel. Let's consider the two horizontal lines (the ones with left - right arrows) as the two lines cut by the vertical transversal (the up - down arrow line). ∠5 is at the upper intersection, outside the two horizontal lines? Wait, no, ∠5 is between the upper vertical and upper horizontal (right), so it's inside the "angle" of the upper intersection. Wait, maybe the problem is considering the two sets of perpendicular lines (each intersection has a horizontal and vertical line) as two parallel lines (horizontal lines are parallel, vertical lines are parallel). So, for the horizontal lines (left - right) cut by the vertical transversal (up - down), ∠5 is at the upper intersection, on the right side of the transversal, above the upper horizontal line? No, ∠5 is at the upper intersection, between the upper vertical (up) and upper horizontal (right), so it's in the first quadrant of the upper intersection. The lower intersection has a horizontal (left - right) and vertical (down) line. ∠7 is at the lower intersection, between the lower vertical (down) and lower horizontal (left), which is in the third quadrant of the lower intersection. Now, alternate exterior angles: when two parallel lines are cut by a transversal, alternate exterior angles are equal and lie outside the two parallel lines, on opposite sides of the transversal. The two horizontal lines (left - right) are parallel. ∠5 is above the upper horizontal line? No, ∠5 is at the intersection, so maybe the "lines" are the horizontal lines (left - right) and the transversal is the vertical line (up - down). Then, ∠5 is on the right side of the transversal, above the upper horizontal line (but it's at the intersection, so maybe the exterior is outside the angle formed by the two horizontal lines). Wait, maybe I was wrong about ∠6. Wait, let's list the angles: ∠5 is at upper intersection, right - upper. ∠7 is at lower intersection, left - lower. Let's check the definition again: Alternate exterior angles are two angles that lie outside the two lines (the two parallel lines) and on opposite sides of the transversal. The two lines here are the horizontal lines (left - right arrows), which are parallel. The transversal is the vertical line (up - down arrows). ∠5 is above the upper horizontal line? No, ∠5 is at the intersection, so it's on the "exterior" of the two horizontal lines? Wait, no, the two horizontal lines are parallel, and the region between them is the interior, and outside is above the upper and below the lower. ∠5 is above the upper horizontal line (since it's at the upper intersection, above the upper horizontal line? No, the upper horizontal line is the left - right arrow at the upper intersection, so ∠5 is at the upper intersection, so it's on the upper horizontal line? Wait, maybe the diagram has two sets of perpendicular lines (like two "plus" signs), one above the other, with the vertical lines (up - down) being the same line (a straight vertical line) and the horizontal lines (left - right) being parallel (two horizontal lines, one above the other). So, the vertical line is the transversal cutting the two parallel horizontal lines. ∠5 is at the upper intersection, on the right side of the transversal, above the upper horizontal line (but the upper horizontal line is the one at the upper intersection, so ∠5 is on the upper horizontal line, right side, above? No, ∠5 is between the upper vertical (up) and upper horizontal (right), so it's a right angle? Wait, all angles at each intersection are right angles? Because they are formed by perpendicular lines. Oh! Wait, each intersection has a horizontal and vertical line, so all angles at each intersection are 90 - degree angles. So, ∠5 is a right angle (90 degrees), and we need to find which angle is an alternate exterior angle with ∠5. Alternate exterior angles for parallel lines cut by a transversal: if two parallel lines (horizontal lines) are cut by a transversal (vertical line), then alternate exterior angles are equal. Since all angles at each intersection are right angles, ∠5 is 90 degrees. Now, ∠7: at the lower intersection, between the lower vertical (down) and lower horizontal (left), is also a right angle. Let's check the positions: ∠5 is at the upper intersection, right - upper (exterior to the two horizontal lines? Wait, no, if the two horizontal lines are parallel, the area between them is interior, and outside is above the upper and below the lower. ∠5 is above the upper horizontal line (since it's at the upper intersection, above the upper horizontal line? No, the upper horizontal line is the left - right line at the upper intersection, so ∠5 is on the upper horizontal line, right side, and ∠7 is on the lower horizontal line, left side, below the lower horizontal line? No, this is confusing. Wait, maybe the key is that alternate exterior angles are on opposite sides of the transversal and outside the two lines. The two lines here are the horizontal lines (left - right), and the transversal is the vertical line (up - down). ∠5 is outside the two horizontal lines (above the upper horizontal line) and on the right side of the transversal. ∠7 is outside the two horizontal lines (below the lower horizontal line) and on the left side of the transversal. So, they are alternate exterior angles. Let's check the other angles: ∠1 is at the lower intersection, right - upper, inside the lower intersection, not exterior. ∠4 is right - lower, inside. ∠6 is left - upper, inside. So, only ∠7 is the alternate exterior angle with ∠5.

Answer:

$\angle7$ (the checkbox next to $\angle7$ should be selected)