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opposite the angle ADB is equal to the side DC opposite the angle DBC (Theo. V. Cor.); and, in like manner, the side A D is equal to the side BC; hence the opposite sides of a parallelogram are equal.

Again, since the triangles are equal, the angle A is equal to the angle C (Theo. V. Cor.); and since the two angles D BC, ABD are respectively equal to the two angles A D B, BD C, the angle ABC is equal to the angle AD C.

80. Cor. 1. The diagonal divides a parallelogram into two equal triangles.

81. Cor. 2. The two parallels AD, BC, included between two other parallels, AB, CD, are equal.

THEOREM ХХІІІ.

82. If the opposite sides of a quadrilateral are equal, each to each, the equal sides are parallel, and the figure is a parallelogram.

Let ABCD be a quadrilateral having its opposite sides equal; then will the equal sides be parallel, and the figure be a parallelogram.

D

A

C

B

For, having drawn the diagonal BD, the triangles ABD, BDC have all the sides of the one equal to the corresponding sides of the other; therefore they are equal, and the angle ADB opposite the side A B is equal to DBC opposite CD (Theo. XIII. Sch.); hence the side AD is parallel to BC (Theo. XV.). For a like reason, A B is parallel to CD; therefore the quadrilateral ABCD is a parallelogram.

THEOREM XXIV.

83. 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 AB, CD equal and parallel; then will the other sides also be equal and parallel.

A

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B

Draw the diagonal BD; then, since AB is parallel to CD, and BD meets them, the alternate angles ABD, BDC are equal (Theo. XVII.); moreover, in the two triangles ABD, DBC, 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 (Theo. IV.), and the side A D is equal to BC. Hence the angle ADB is equal to DBC, and consequently A D is parallel to BC (Theo. XV.); therefore the figure ABCD is a parallelogram.

THEOREM XXV.

84. The diagonals of every parallelogram bisect each other, or divide each other into equal parts.

Let ABCD be a parallelogram, and AC, DB its diagonals, intersecting at E; then will A E equal EC, and BE equal E D.

D

C

E

A

B

For, since AB, CD are parallel, and BD meets them, the alternate angles CDE, ABE are equal (Theo. XVII.); and since AC meets the same parallels, the alternate angles BAE, ECD are also equal; and the sides AB, CD are equal (Theo. XXII.). Hence the triangles ABE, CDE have two angles and the included side in the one equal to two angles and the included side in the other, each to each; hence the two triangles are equal (Theo. V.); therefore the side A E opposite the angle ABE is equal to CE opposite CDE; hence, also, the sides BE, DE opposite the other equal angles are equal.

85. Scholium. In the case of a rhombus, the sides A B, BC being equal, the triangles AEB, 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 AEB, BEC are equal. Therefore the diagonals of a rhombus bisect each other at right angles.

EXERCISES

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FOR ORIGINAL THOUGHT, ON REVIEW.

86. 1. If the opposite angles formed by four lines meeting at a point are equal, these lines form but two straight lines. 2. If the equal sides of an isosceles triangle are produced, the two exterior angles formed with the base will be equal.

3. The line bisecting the vertical angle of an isosceles triangle bisects the base at right angles.

4. Every equilateral triangle is also equiangular.

5. Any side of a triangle is less than the sum of the other

two.

6. The difference between any two sides of a triangle is less than the other side.

7. The side in a right-angled triangle opposite the right angle is the longest.

8. In every equilateral triangle each of the angles will be equal to two-thirds of one right angle.

9. Straight lines which are parallel to the same line are parallel to each other.

10. Lines joining the corresponding extremities of two equal and parallel straight lines, are themselves equal and parallel, and the figure formed is a parallelogram.

11. If, in the sides of a square, at equal distances from the four angles, points be taken, one in each side, the straight lines joining these points will form a square.

12. If from the middle point of a straight line a perpendicular be drawn, any point in the perpendicular will be equally distant from the extremities of the line.

BOOK II.

RATIO AND PROPORTION.

DEFINITIONS.

87. 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.

88. 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 by A: B.

or

89. 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.

Ratios of magnitudes may be expressed by numbers either exactly, or approximately.

90. 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.

91. 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. Magnitudes of the same kind which have a common measure are said to be commensurable; and those which have no common measure are said to be incommensurable.

92. 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. 93. A Compound Ratio is the product of two or more ratios.

or

Thus the ratio compounded of A: B and C :D is

AXC

BXD

94. A Proportion is an equality of ratios.

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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 each

other, when written A : B = X : Y, or

portion.

A

X

B

Y'

form a pro

95. Proportion is written not only with the sign =, but, more often, with the sign :: between the ratios.

Thus, A: B :: X: Y, expresses a proportion, and is read, The ratio of A to B is equal to the ratio of X to Y; or, A is to Bas X is to Y.

96. The first and third terms of a proportion are called the Antecedents; the second and fourth, the Consequents. The first and fourth are also called the Extremes, and the second and third the Means.

Thus, in the proportion A: B: : C :: D, A and Care the

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