Let AB, CD be equally inclined to EF; and let GH be any other transversal. It shall be proved that they are equally inclined to GH.
Join EH.
Because ∠s of Triangle EFH together = 2 rt ∠s, and likewise those of Triangle EGH, [(λ).
∴ angles of Figure FG together = 4 rt angles;
also, by hypothesis, ∠s GEF, EFH together = 2 rt ∠s;
∴ remaining ∠s EGH, GHF together = 2 rt ∠s;
∴ AB, CD are equally inclined to GH.
Therefore a Pair of Lines, &c. Q. E. D.
Contranominal of (α). II. 2.
A Pair of Lines, which make with a third Line two interior angles, on one side of it, together less than two right angles, will meet on that side if produced.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/3/31/Euclid_and_His_Modern_Rivals_Page259.png/350px-Euclid_and_His_Modern_Rivals_Page259.png)
Let ABC, DEF be two Triangles such that ∠s, A, D are equal, and DE, DF equimultiples of AB, AC.
From DE cut off successive parts equal to AB; and let the points of section be G, H. At G, H make ∠s equal to ∠E.
Then the Lines, so drawn, are separational from EF and from one another; [Euc. I. 28.
∴ these Lines meet DF between D and F; call these points K, L.
At G, H make ∠s equal to ∠D.
Then the Lines, so drawn, are separational from DF;
∴ they respectively meet HL between H and L, and EF between E and F; call these points M, N.
S 2