Euclid's Elements: Book I: Proposition 28

Proposition 28: If a straight line falling on two straight lines make the exterior angle equal to the interior and opposite angle on the same side, or the interior angles on the same side equal to two right angles, the straight lines will be parallel to one another.

For let the straight line EF falling on the two straight lines AB, CD make the exterior angle EGB equal to the interior and opposite angle GHD, or the interior angles on the same side, namely BGH, GHD, equal to two right angles;

I say that AB is parallel to CD.

For, since the angle EGB is equal to the angle GHD, while the angle EGB is equal to the angle AGH, I. 15 the angle AGH is also equal to the angle GHD; and they are alternate; therefore AB is parallel to CD. I. 27

Again, since the angles BGH, GHD are equal to two right angles, and the angles AGH, BGH are also equal to two right angles, I. 13 the angles AGH, BGH are equal to the angles BGH, GHD.

Let the angle BGH be subtracted from each; therefore the remaining angle AGH is equal to the remaining angle GHD; and they are alternate;

Therefore AB is parallel to CD. I. 27

Therefore etc.

Euclid's Elements: Book I
< Proposition 27 | Proposition 28 | Proposition 29 >

```                       E
/
d   /  a
A ------------------G------------- B
c   /  b
/
/
h   /   e
C -------------H------------------ D
g   /   f
/
F
```

Claim:
If a = e, then AB parallel to CD.

Proof:
a = e is given.
a = c by proposition 15.
Hence e = c.
AB parallel to CD by proposition 27.
This completes the proof.

This is the last of the 28 propositions that do not depend on the parallel postulate in Euclid's Elements. Hence it is one of the postulates that holds for non-euclidian geometries also.

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