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Proposition 27

If a straight line falling on two straight lines makes the alternate angles equal to one another, then the straight lines are parallel to one another.
Let the straight line EF falling on the two straight lines AB and CD make the alternate angles AEF and EFD equal to one another. I say that AB is parallel to CD.
If not, AB and CD when produced meet either in the direction of B and D or towards A and C. Let them be produced and meet, in the direction of B and D, at G. |
Then, in the triangle GEF, the exterior angle AEF equals the interior and opposite angle EFG, which is impossible. | I.16
Therefore AB and CD when produced do not meet in the direction of B and D. Similarly it can be proved that neither do they meet towards A and C.
But straight lines which do not meet in either direction are parallel. Therefore AB is parallel to CD. | I.Def.23
Therefore if a straight line falling on two straight lines makes the alternate angles equal to one another, then the straight lines are parallel to one another.
Q.E.D.

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There is implicitly assumed an ambient plane. The term "alternate angles" doesn't have a meaning unless the lines all lie in a plane.

Note that Euclid does not consider two other possible ways that the two lines could meet, namely, in the directions A and D or toward B and C.

#### About logical inverses

Although this is the first proposition about parallel lines, it does not require the parallel postulate Post.5 as an assumption. This proposition I.27 and the parallel postulate can be made to look more similar if they are reworded (with the help of I.13).

Proposition 1.27.
  If a straight line falls on two straight lines, then if the alternate angles are equal, then the straight lines do not meet.

Post.5.
  If a straight line falls on two straight lines, then if the alternate angles are not equal, then the straight lines meet [on a certain side of the line].

If the remark about the side is dropped, then the conclusions are logical inverses of each other, and the logical inverse of a statement is logically equivalent to the converse.

This little table summarizes the logical relations between similarly looking statements.

Statement | If P then Q.
Converse | If Q then P. | Not logically equivalent to the statement.
Contrapositive | If not Q then not P. | Logically equivalent to the statement.
Inverse | If not P then not Q. | Logically equivalent to the converse.

Although the contrapositive is logically equivalent to the statement, Euclid always proves the contrapositive separately using a proof by contradiction and the original statement. Similarly, the inverse is proved using the converse. Sometimes all four statements appear in separate propositions as in propositions X.5 through X.8. Other times the four appear as four statements in one proposition as in X.9. More often than not, however, the contrapositive and inverse make no appearance, and, of course, the converse only appears when it can be proved.

#### Use of Proposition 27

At this point, parallel lines have yet to be constructed. That occurs in proposition I.31 which uses this proposition to verify that lines constructed there are parallel. This proposition is also used in the next one and in I.33.

Book I Introduction - Proposition 26 - Proposition 28.

© 1996, 1997
D.E.Joyce
Clark University