Prop. 33. The straight lines which join the extremities of two equal and parallel straight lines towards the same parts are themselves equal and parallel; and
Prop. 34. The opposite sides and angles of a parallelogram are equal to one another, and the diameter (diagonal) bisects the parallelogram, that is, divides it into two equal parts.
§ 17. The rest of the first book relates to areas of figures.
The theory is made to depend upon the theorems—
Prop. 35. Parallelograms on the same base and between the same parallels are equal to one another; and
Prop. 36. Parallelograms on equal bases and between the same parallels are equal to one another.
As each parallelogram is bisected by a diagonal, the last theorems hold also if the word parallelogram be replaced by “triangle,” as is done in Props. 37 and 38.
It is to be remarked that Euclid proves these propositions only in the case when the parallelograms or triangles have their bases in the same straight line.
The theorems converse to the last form the contents of the next three propositions, viz.: Props, 40 and 41.—Equal triangles, on the same or on equal bases, in the same straight line, and on the same side of it, are between the same parallels.
That the two cases here stated are given by Euclid in two separate propositions proved separately is characteristic of his method.