1

Proposition 1.28 states:

If a straight line falling on two straight lines makes the sum of the interior angles on the same side equal to two right angles, then the straight lines are parallel to one another.

Euclid has given a somewhat long proof of this but I believe it is a direct consequence of his fifth 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.

What am I missing?

Siddhartha
  • 1,770
  • 18
  • 38
  • 2
    The Fifth Postulate says what happens if the angle sum is less than two right angles. It makes no claims about what happens in the case of equality. – Blue Dec 21 '19 at 13:09
  • If the sum is less than two right angles, the lines meet on one side. Consequently, if the sum is greater than two right angles, the lines meet on the other side. Thus, the lines can be parallel if and only if the sum equals two right angles. – Siddhartha Dec 21 '19 at 13:24
  • 4
    There is room in your reasoning for the possibility that they meet even if the sum of the angles is exactly two right angles. What you have shown (by way of contrapositive) is that if the lines are parallel then the angle sum equals two right angles. – Karl Kroningfeld Dec 21 '19 at 13:28
  • 1
    @Thomas In your comment, the first two sentences are correct, but in the last sentence, you wrote "if and only if" where the previous sentences only established the "only if" direction. – Andreas Blass Dec 21 '19 at 16:08

0 Answers0