DEFINITIONS. XXVII. A PARALLELOGRAM is a four-sided figure whose oppo site sides are parallel. D For brevity we often designate a parallelogram by two letters only, which mark opposite angles. Thus we call the figure in the margin the parallelogram AC. XXVIII. A RECTANGLE is a parallelogram, having one of its angles a right angle. XXIX. A RHOMBUS is a parallelogram, having its sides equal. XXX. A SQUARE is a parallelogram, having its sides equal and one of its angles a right angle. XXXI. A TRAPEZIUM is a four-sided figure of which two sides only are parallel. XXXII. A DIAGONAL of a four-sided figure is the straight line joining two of the opposite angular points. XXXIII. The ALTITUDE of a Parallelogram is the perpendicular distance of one of its sides from the side opposite, regarded as the Base. The altitude of a triangle is the perpendicular dis tance of one of its angular points from the side opposite, regarded as the base. Thus if ABCD be a parallelogram, and AE a perpendicular let fall from A to CD, AE is the altitude of the parallelogram, and also of the triangle ACD. B D E If a perpendicular be let fall from B to DC produced, meeting DC in F, BF is the altitude of the parallelogram. EXERCISES. Prove from the definitions just given the following theo rems: 1. All the angles of a Square are right angles. 2. All the angles of a Rectangle are right angles. 3. The diagonals of a square make with each of the sides an angle equal to half a right angle. 4. If two straight lines bisect each other, the lines joining their extremities will form a parallelogram. 5. Straight lines bisecting two adjacent angles of a parallelogram intersect at right angles. 6. If the straight lines joining two opposite angular points of a parallelogram bisect the angles, the parallelogram is a rhombus. 7. If the opposite angles of a quadrilateral be equal, the quadrilateral is a parallelogram. 8. If two opposite sides of a quadrilateral figure be equal to one another, and the two remaining sides be also equal to one another, the figure is a parallelogram. 9. If one angle of a rhombus be equal to two-thirds of two right angles, the diagonal drawn from that angular point divides the rhombus into two equilateral triangles. PROPOSITION XXXIV. THEOREM. The opposite sides and angles of a parallelogram are equal to one another, and the diagonal bisects it. and and Let ABDC be a □, and BC a diagonal of the. Then must AB DC and AC=DB, ▲ BAC= ▲ CDB, and ▲ ABD= 4 ACD ▲ ABC= ▲ DCB. For AB is to CD, and BC meets them, .. LABC alternate 2 DCB; I. 29. and AC is to BD, and BC meets them, .. LACB alternate DBC. Then in As ABC, DCB, :: LABC= L DCB, and 4 ACB= L DBC, I. 29. and BC is common, a side adjacent to the equals in each; .. AB=DC, and AC=DB, and L BAC and AABC= ▲ DCB. LCDB, I. B. Also L ABC= L BCD, and ▲ CBD= L ACB, .. 48 ABC, CBD together= 48 BCD, ACB together, that is, ABD= ▲ ACD. I. 4. Q. E. D. Ex. 1. Shew that the diagonals of a parallelogram bisect each other. Ex. 2. Shew that the diagonals of a rectangle are equal. Ex. 3. Prove that the four triangles, into which a parallelogram is divided by its diagonals, are equal to each other. PROPOSITION XXXV. THEOREM. Parallelograms on the same base and between the same parallels are equal. Let the s ABCD, EBCF be on the same base BC, and between the same s AF, BC. CASE I. If there be a space between the sides AD, EF. Now ABCD with ▲ FDC=figure ABCF; and EBCF with ▲ EAB=figure ABCF; ABCD with ▲ FDC=□EBCF with ▲ EAB; CASE II. thus: If the sides AD, EF overlap one another, ED F the same method of proof applies. CASE III. If the sides opposite to BC be terminated in the same point D, thus: B the same method of proof is applicable, Each of the s is double of ▲ BDC; ABCD=DBCF. I. 34. Q. E.D. Parallelograms on equal bases, and between the same parallels, are equal to one another. Let the s ABCD, EFGH be on equal bases BC, FG, and between the same s AH, BG. they are on the same base BC and between the same ||s; and EBCH=□EFGH, I. 35. they are on the same base EH and between the same s ; |