Euclid, Elements
Thomas L. Heath, Sir Thomas Little Heath, Ed.

But the three angles LOM, MON, NOL are equal to four right angles; therefore the angles ABC, DEF, GHK are equal to four right angles.

But they are also, by hypothesis, less than four right angles: which is absurd.

Therefore AB is not equal to LO.

I say next that neither is AB less than LO.

For, if possible, let it be so, and let OP be made equal to AB, and OQ equal to BC, and let PQ be joined.

Then, since AB is equal to BC, OP is also equal to OQ, so that the remainder LP is equal to QM.

Therefore LM is parallel to PQ, [VI. 2] and LMO is equiangular with PQO; [I. 29] therefore, as OL is to LM, so is OP to PQ; [VI. 4] and alternately, as LO is to OP, so is LM to PQ. [V. 16]

But LO is greater than OP; therefore LM is also greater than PQ.

But LM was made equal to AC; therefore AC is also greater than PQ.

Since, then, the two sides AB, BC are equal to the two sides PO, OQ, and the base AC is greater than the base PQ, therefore the angle ABC is greater than the angle POQ. [I. 25]

Similarly we can prove that the angle DEF is also greater than the angle MON, and the angle GHK greater than the angle NOL.

Therefore the three angles ABC, DEF, GHK are greater than the three angles LOM, MON, NOL.

But, by hypothesis, the angles ABC, DEF, GHK are less than four right angles; therefore the angles LOM, MON, NOL are much less than four right angles.