DEFINITIONS. XXVII. A PARALLELOGRAM is a four-sided figure whose opposite 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. Hence by I. 29, all the angles of a rectangle are right angles. 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. Hence, by I. 29, all the angles of a square are right angles. 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 distance 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. If a perpendicular be let fall from B to DC produced, meeting DC in F, BF is the altitude of the parallelogram. EXERCISES. Prove the following theorems: 1. The diagonals of a square make with each of the sides an angle equal to half a right angle. 2. If two straight lines bisect each other, the lines joining their extremities will form a parallelogram. 3. Straight lines bisecting two adjacent angles of a parallelogram intersect at right angles. 4. If the straight lines joining two opposite angular points of a parallelogram bisect the angles, the parallelogram is a rhombus. 5. If the opposite angles of a quadrilateral be equal, the quadrilateral is a parallelogram. 6. 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. 7. 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. D Let ABDC be a □, and BC a diagonal of the O. and and ▲ BAC= 2 CDB, and ▲ ABD= 4 ACD ▲ ABC ADCB. For AB is to CD, and BC meets them, .. ABC alternate DCB; and AC is to BD, and BC meets them, I. 29. I. 29. ** 2 ABC= 2 DCB, and ▲ ACB= 2 DBC, and BC is common, a side adjacent to the equal ▲ s in each; .. AB=DC, and AC=DB, and ▲ BAC= 2 CDB, Also and ▲ ABC= ▲ DCB. ABC= 4 DCB, and ▲ DBC= ▲ ACB, .. Ls ABC, DBC togethers DCB, ACB together, I. B. that is, = L ABD LACD. 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. Parallelograms on the same base and between the same parallels are equal. DE Let the Os ABCD, EBCF be on the same base BC. and between the same s AF, BC. Then must ☐ ABCD=□ EBCF. CASE I. If AD, EF have no point common to both, Then in the As FDC, EAB, ABCD with ▲ FDC=□ EBCF with ▲ EAB; CASE II. If the sides AD, EF overlap one another, CASE III. If the sides opposite to BC be terminated in the same point D, W B the same method of proof is applicable, Each of the Os is double of a BDC; .. ABCD=□ DBCF. I. 34. Q. E. D. Parallelograms on equal bases, and between the same parallels, are equal to one another. Let the Os ABCD, EFGH be on equal bases BC, FG, and between the same Is 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; ..ABCDEFGH. Q. E. D. |