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.
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.
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