Rebecca Villa

2022-07-01

The Corresponding Angles Postulate states that, when two parallel lines are cut by a transversal , the resulting corresponding angles are congruent. Is this a direct obvious consequence of Euclid's fifth postulate? I think so, because the fifth postulate is basically saying that through a point there is only one line passing parallel to another; am I right? If not, how can you be sure your geometry is consistent having defined parallel lines as those that never meet?

Jordin Church

Expert

"because the fifth postulate is basically saying that through a point there is only one line passing parallel to another" well, technically that is not Euclid's 5th postulate.

Euclid's 5 postulate is:

Euclid's 5 postulate: That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

This is equivalent to

Exactly one Parallel line: Given a line and a point not one the line then there is exactly one line through the point that does not intersect the original line.

Euclid's 5th implies exactly one parallel.

Draw a perpendicular from the line to the point. Draw a line through the point. If measure the angles of from the line to the perpedicular. If the angle are not two right triangles the the two lines will might one the side that is less than a right and the lines are not parallel. If the angles are perpendicular, the we consider if the lines intersect. If they intersect they must interesect one one side (lines don't intersect at two points). But if they intersect on one side they must by symmettry intersect on the other. So that line is parallel. Remains to show that there is exactly one line through the point that is perpendicular to the perpendicular. And ... argh... i'm too tired to work that out.

Exactly one parallel line $\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}$ Euclid's 5th.

Consider the line and the one parallel line through the point. Measure the interior angles made by a transversal. If the angles do not measure 180, we can construct a second line through the point with where the interior angles by the same transversal are complimentary. This will in, in essence just be the mirror image. By similarity either both lines will be parallel or neither will. But as there is exactly one this is a contradiction. So parallel lines must have angles that measure 180 and lines that don't intersect. If they intersect on the larger angle side we can construct a small angle within it and that must intersect and by symmetry the must intersect on the small side as well. As they can only intersect once they intersect on the smaller side.

Do you have a similar question?