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112. If two opposite sides of a quadrilateral are equal and parallel, the other sides are also equal and parallel, and the figure is a parallelogram.
Let ABCD be a quadrilateral, having the sides A B, C D equal and parallel; then will the other sides. also be equal and parallel.
Draw the diagonal BD; then, since A
A B is parallel to CD, and BD meets them, the alternate angles A BD, BDC are equal (Prop. XXII.); moreover, in the two triangles A BD, D B C, the side BD is common; therefore, two sides and the included angle in the one are equal to two sides and the included angle in the other, each to each; hence these triangles are equal (Prop. V.), and the side A D is equal to BC. Hence the angle A D B is equal to D B C, and consequently A D is parallel to BC (Prop. XX.); therefore the figure ABCD is a parallelogram.
PROPOSITION XXXIV. - THEOREM.
113. The diagonals of every parallelogram bisect each other.
Let ABCD be a parallelogram, and A C, DB its diagonals, intersecting at E; then will A E equal E C, and B E equal ED.
For, since A B, CD are parallel, and BD meets them, the alternate angles CDE, ABE are equal (Prop. XXII.); and since A C meets the same parallels, the alternate angles BAE, ECD are also equal; and the sides AB, CD are equal (Prop. XXXI.). Hence the triangles ABE, CDE have two angles and the in
cluded side in the one equal to two angles and the included side in the other, each to each; hence the two triangles are equal (Prop. VI.); therefore the side A E opposite the angle A B E is equal to CE opposite CDE; hence, also, the sides BE, DE opposite the other equal angles are equal.
114. Scholium. In the case of a rhombus, the sides A B, BC being equal, the triangles A E B, EBC have all the sides. of the one equal to the corresponding sides of the other, and are, therefore, equal; whence it follows that the angles A EB, BEC are equal. Therefore the diagonals of a rhombus bisect each other at right angles.
PROPOSITION XXXV. - THEOREM.
115. If the diagonals of a quadrilateral bisect each other, the figure is a parallelogram.
Let ABCD be a quadrilateral, and D AC, D B its diagonals intersecting at E; then will the figure be a parallelogram.
For, in the two triangles ABE, CDE, the two sides AE, EB and the included A angle in the one are equal to the two sides CE, ED and the included angle in the other; hence the triangles are equal, and the side AB is equal to the side CD (Prop. V. Cor.). For a like reason, A D is equal to C B ; therefore the quadrilateral is a parallelogram (Prop. XXXII.).
RATIO AND PROPORTION.
116. RATIO is the relation, in respect to quantity, which one magnitude bears to another of the same kind; and is the quotient arising from dividing the first by the second. A ratio may be written in the form of a fraction, or with the sign :
Thus the ratio of A to B may be expressed either by
or by A: B.
117. The two magnitudes necessary to form a ratio are called the TERMS of the ratio. The first term is called the ANTECEDENT, and the last, the CONSEQUENT.
118. Ratios of magnitudes may be expressed by numbers, either exactly, or approximately.
This may be illustrated by the operation of finding the numerical ratio of two straight lines, A B, CD.
From the greater line AB cut off a part equal to the less CD, as many times as possible; for example, twice, with the remainder B E.
From the line CD cut off a part equal to the remainder BE as many times as possible; once, for example, with the remainder DF.
From the first remainder BE, cut off a part equal to the second D F, as many times as possible; once, for example, with the remainder BG.
Proceed thus till a remainder arises, which is exactly contained a certain number of times in the preceding one.
Then this last remainder will be the common measure of the proposed lines; and, regarding it as unity, we shall easily find the values of the preceding remainders; and, at last, those of the two proposed lines, and hence their ratio in numbers.
Suppose, for instance, we find GB to be contained exactly twice in FD; BG will be the common measure of the two proposed lines. Let BG equal 1; then will FD equal 2. But EB contains FD once, plus GB; therefore we have EB equal to 3. CD contains E B once, plus FD; therefore we have CD equal to 5. AB contains CD twice, plus EB; therefore we have A B equal to 13. Hence the ratio of the two lines is that of 13 to 5. If the line CD were taken for unity, the line AB would be; if AB were taken for unity, CD would be.
It is possible that, however far the operation be continued, no remainder may be found which shall be contained an exact number of times in the preceding one. In that case there can be obtained only an approximate ratio, expressed in numbers, more or less exact, according as the operation is more or less extended.
119. When the greater of two magnitudes contains the less a certain number of times without having a remainder, it is called a MULTIPLE of the less; and the less is then called a SUBMULTIPLE, or measure of the greater.
Thus, 6 is a multiple of 2; 2 and 3 are submultiples, or measures, of 6.
120. EQUIMULTIPLES, or LIKE MULTIPLES, are those which contain their respective submultiples the same number of
times; and EQUISUBMULTIPLES, or LIKE SUBMULTIPLES, are those contained in their respective multiples the same number of times.
Thus 4 and 5 are like submultiples of 8 and 10; 8 and 10 are like multiples of 4 and 5.
121. COMMENSURABLE magnitudes are magnitudes of the same kind, which have a common measure, and whose ratio therefore may be exactly expressed in numbers.
122. INCOMMENSURABLE magnitudes are magnitudes of the same kind, which have no common measure, and whose ratio, therefore, cannot be exactly expressed in numbers.
123. A DIRECT ratio is the quotient of the antecedent by the consequent; an INVERSE ratio, or RECIPROCAL ratio, is the quotient of the consequent by the antecedent, or the reciprocal of the direct ratio.
Thus the direct ratio of a line 6 feet long to a line 2 feet long is or 3; and the inverse ratio of a line 6 feet long to a line 2 feet long is or, which is the same as the reciprocal of 3, the direct ratio of 6 to 2.
The word ratio when used alone means the direct ratio. 124. A COMPOUND ratio is the product of two or more ratios.
Thus the ratio compounded of A: B and C D is A C AX C X or B D' BX D
125. A PROPORTION is an equality of ratios.
Four magnitudes are in proportion, when the ratio of the first to the second is the same as that of the third to the fourth.
Thus, the ratios of A: B and X: Y, being equal to A
each other, when written A: B = X: Y, or B form a proportion.