Euclid's Elements: Book I: Proposition 29

Proposition 29: A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles.

For let the straight line EF fall on the parallel straight lines AB, CD;

I say that it makes the alternate angles AGH, GHD equal, the exterior angle EGB equal to the interior and opposite angle GHD, and the interior angles on the same side, namely BGH, GHD, equal to two right angles.

For, if the angle AGH is unequal to the angle GHD, one of them is greater.

Let the angle AGH be greater.

Let the angle BGH be added to each; therefore the angles AGH, BGH are greater than the angles BGH, GHD.

But the angles AGH, BGH are equal to two right angles; I. 13 therefore the angles BGH, GHD are less than two right angles.

But straight lines produced indefinitely from angles less than two right angles meet; Post. 5 See Note

therefore AB, CD, if produced indefinitely, will meet; but they do not meet, because they are by hypothesis parallel.

Therefore the angle AGH is not unequal to the angle GHD, and is therefore equal to it.

Again, the angle AGH is equal to the angle EGB; I. 15 therefore the angle EGB is also equal to the angle GHD. C.N. 1

Let the angle BGH be added to each; therefore the angles EGB, BGH are equal to the angles BGH, GHD. C.N. 2

But the angles EGB, BGH are equal to two right angles; I. 13

Therefore the angles BGH, GHD are also equal to two right angles.

Therefore etc.

This proposition is very important as contains the first reference to postulate 5. This postulate has a similar effect on the structure of Euclid's Elements to that of Elements I.4. Removing this postulate enabled geometers to "discover" Non-Euclidean Geometry.

Euclid's Elements: Book I
< Proposition 28 | Proposition 29 | Proposition 30 >

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

Claim:
If AB parallel to CD, then c = e, a = e, and b + e = 180

Suppose c not equal to e.
Wlog, let e < c.
b + e < b + c
b + e < 180 (proposition 13 on supplementary angles)
Hence AB, CD meet (parallel postulate).
Contradiction with hypothesis that AB, CD are parallel.
Therefore c = e.
c = a (proposition 15)
e = a
b + e = b + a
b + e = 180 (proposition 13)
This completes the proof.
```
This proposition is the converse of the previous two propositions. All propositions hereupon in Book I rely on the parallel postulate.