...is derived from, or is the total result of, the previous deductions (i) that ' Parallelograms upon equal bases, and between the same parallels, are equal to one another,' (2) that 'Triangles formed by the diagonal of a parallelogram are each of them equal to half the parallelogram'...

...ABC is equal to the area of the triangle DBC (Bk. IV. Prop. i, Cor.). PROPOSITION 7. Triangles upon equal bases and between the same parallels are equal to one another in area. Let ABC and DEF be two triangles upon the equal bases BC and EF, and between the same parallels...

...and 2 doors, each 9 feet by four, with paper I £ yards wide, at 5jd. a yard. 4. Parallelograms upon equal bases and between the same parallels are equal to one another. Prove this ; and modify the enunciation, so as to extend it to the case where the parallelograms are...

...straight lines, and the lines AD, BC intersect in E, then the triangles AEC, BED are equal. Triangles upon equal bases, and between the same parallels, are equal to one another. Let AS ABC, DEF be on equal bases, BC, EF and between the same IIs BF, AD. Then must A ABC= A DEF....

...ote. — This illustrates the important geometrical truth, that — Parallelograms on the same base and between the same parallels are equal to one another; that is, equal in area. Fic.34 AEFD and ABCD are on the same base AD and between the same parallels AD and EC....

...reason thus : Each of the O« is double of &BDC; 1.34. QED PROPOSITION XXXVI. THEOREM. Parallelograms on equal bases, and between the same parallels, are equal to one another. A. J) M !_-. JT Let the Os A BCD, EFGH be on equal bases BC, FG, and between the same II s AH, BG....

...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...

...bisects it ; and DEC is the half of DBCF ; therefore ABC is equal to DEC. XXXVIII. — Triangles upon equal bases, and between the same parallels, are equal to one another. Let the triangles ABC, DEF, be upon equal bases BC, EF, and between the same parallels BF, AD. Produce...

...CB ; shew that the triangle AOD is equal to the triangle BOC. PROPOSITION 38. THEOREM. Triangles on equal bases, and between the same parallels, are equal to one another. . Let the triangles ABC, DEF be on equal bases 1 • BC EF and between the same parallels Bf, AD :...

...consecutive angles of a quadrilateral is half the sum of the two remaining angles. 2. Parallelograms -on equal bases, and between the same parallels, are equal to one another. 3. Divide a given straight line into two parts, so that the rectangle contained by the whole and one...