...quantity. This, which is a very useful property, is usually cited in the words, " Parallelograms upon the same or equal bases, and between the same parallels, are equal to each other." But there is some objection to the words "between the same parallels," inasmuch as,...

...quantity. This, which is a very useful property, is usually cited in the words, " Parallelograms upon the same or equal bases, and between the same parallels, are equal to each other." But there is some objection to the words " between the same parallels," inasmuch as,...

...proved, that, ABDC being a parallelogram, AB is equal to CD, and AC equal to BD. L. Аи parallelograms on the same or equal bases, and between the same parallels, are equal to nach oí/кг ; that is, if BD and GH be equal, and the lints BH, AF, be parallel, tkt parallelograms...

...has been proved that ABCD being a parallelogram, AB = CD and AD =BC. THEO. XII. All parallelograms on the same or equal bases and between the same parallels are equal to one another, that is if BD — GH, and the lines BH and AF are parallel, then the paralellogram...

...things must themselves be equal; .'. the triangle BCD = the triangle FGH ' that is, all triangles upon the same, or equal bases, and ' between the same parallels', are equal to one another. * ' Parallelogram ' being a long word, when it occurs frequently, it is allowable to...

...which is half of AB EF, is equal to half ABCD, or the parallelogram is double of the triangle. COR. Triangles on the same or equal bases, and between the same parallels are equal. For parallelograms under these two conditions are, by No. 133, and Cor., equal, and the triangles being...

...problems of this chapter. b. Parallelograms (squares, rectangles, and rhomboid figures) or triangles upon the same or equal bases, and between the same parallels, are equal in area (Eviclid, Tk. I., Defs. 35-38). a. The squares on the base and perpendicular of a right-angled...

...equal segments and sectors of circle — and, also, the equality of the areas of parallelograms and triangles on the same or equal bases and between the same parallels. Thus, in the first four chapters, almost the whole substance, theorems and problems, of Euclid's first,...

...С D = area ABGH, and area EFGH = area ABGH ; Consequently area A В С D = area EFGH. THEOREM XLV. Triangles on the same or equal bases and between the same parallels have equal areas. Each of the triangles is half any parallelogram on the same or an equal base and...

...of a figure depends both upon its shape and upon its perimeter. 2. Parallelograms and triangles upon the same or equal bases, and between the same parallels, are equal in area (Euclid i. 35-38). 3. If a parallelogram and a triangle be upon the same or equal bases, and...