Euclid was working in a context of classical geometric construction with compass and straightedge. If you have a line and a point not on the line, you can clas

Euclid was working in a context of classical geometric construction with compass and straightedge. If you have a line and a point not on the line, you can classically construct a perpendicular through the point. Use a compass centered on the given point, open to an arbitrary radius greater than the distance between the given point and the line. Draw a circle, which will intersect the line at two points A and B. Now draw a second circle centered on A and passing through B, and a third circle centered on B and passing through A. The second and third circles intersect in two points, C and D, one on each side of the given line. Draw a new line through C and D. This line is perpendicular to the original line and forms four right angles with it. A way to check whether a given angle is right is to pick an arbitrary point on one of the lines and draw a circle centered there and passing through an arbitrary point on the other line. The circle will then pass through a second point on the other line. You can construct the perpendicular bisector of the segment formed by the two points in which the circle intersects the second line, and see whether the bisector is identical to the first line. (Although I'm not sure that's the right classical solution because I don't recall whether "noticing whether two constructed lines are identical" or "noticing whether a constructed line intersects a given point" are allowed tasks in classical construction.)