Let DB be joined, and let the parallelogram FH be constructed equal to the triangle ABD , in the angle HKF which is equal to E ; [[I. 42](elem.1.42)] let the parallelogram GM equal to the triangle DBC be applied to the straight line GH , in the angle GHM which is equal to E . [[I. 44](elem.1.44)]

Then, since the angle E is equal to each of the angles HKF , GHM , the angle HKF is also equal to the angle GHM . [[C.N. 1](elem.1.c.n.1)]

Let the angle KHG be added to each; therefore the angles FKH , KHG are equal to the angles KHG , GHM .

But the angles FKH , KHG are equal to two right angles; [[I. 29](elem.1.29)] therefore the angles KHG , GHM are also equal to two right angles.

Thus, with a straight line GH , and at the point H on it, two straight lines KH , HM not lying on the same side make the adjacent angles equal to two right angles; therefore KH is in a straight line with HM . [[I. 14](elem.1.14)]

And, since the straight line HG falls upon the parallels KM , FG , the alternate angles MHG , HGF are equal to one another. [[I. 29](elem.1.29)]

Let the angle HGL be added to each; therefore the angles MHG , HGL are equal to the angles HGF , HGL . [[C.N. 2](elem.1.c.n.2)]

But the angles MHG , HGL are equal to two right angles; [[I. 29](elem.1.29)] therefore the angles HGF , HGL are also equal to two right angles. [[C.N. 1](elem.1.c.n.1)] Therefore FG is in a straight line with GL . [[I. 14](elem.1.14)]

And, since FK is equal and parallel to HG , [[I. 34](elem.1.34)] and HG to ML also,
KF is also equal and parallel to ML ; [[C.N. 1](elem.1.c.n.1);[ I. 30](elem.1.30)] and the straight lines KM , FL join them (at their extremities); therefore KM , FL are also equal and parallel. [[I. 33](elem.1.33)] Therefore KFLM is a parallelogram.

And, since the triangle ABD is equal to the parallelogram FH , and DBC to GM ,
the whole rectilineal figure ABCD is equal to the whole parallelogram KFLM .

Therefore the parallelogram KFLM has been constructed equal to the given rectilineal figure ABCD , in the angle FKM which is equal to the given angle E .

Q. E. F.

rectilineal

simply, without figure,

εὐθύγραμμον being here used as a substantive, like the similarly formed παραλληλόγραμμον .