Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση
[ocr errors]

THEOREM.

79. The sum of all the interior angles of a polygon is equal to as

many times two right angles, as there are units in the number of sides, diminished by two.

E

Let ABCDEFG be the proposed polygon. If from the vertex of any one angle A, diago- B nals AC, AD, AE, AF, be drawn to the vertices of all the opposite angles, it is plain that A the polygon will be divided into five triangles, if it has seven sides; into six triangles, if it has eight; and, in general, into as many triangles, less two, as the polygon has sides; for those triangles may be considered as having the point A for a common vertex, and for bases, the several sides of the polygon, excepting the two sides which form the angle A. It is evident, also, that the sum of all the angles in those triangles does not differ from the sum of all the angles in the polygon: hence the latter sum is equal to as many times two right angles as there are triangles in the figure; in other words, as there are units in the number of sides diminished by two.

80. Cor. 1. The sum of the angles in a quadrilateral is equal to two right angles multiplied by 4-2, which amounts to four right angles: hence if all the angles of a quadrilateral are equal, each of them will be a right angle; a conclusion which sanctions our seventeenth Definition, where the four angles of a quadrilateral are asserted to be right, in the case of the rectangle and the square.

81. Cor. 2. The sum of the angles of a pentagon is equal to two right angles multiplied by 5-2, which amounts to six right angles: hence when a pentagon is equiangular, each angle is equal to the fifth part of six right angles, or to of one right angle.

82. Cor. 3. The sum of the angles of a hexagon is equal to 2× (6--2), or eight right angles: hence in the equiangular hexagon, each angle is the sixth part of eight right angles, or of one.

83. Scholium. When this proposition is applied to polygons which have re-entrant angles, each re-entrant angle must be regarded as greater than two right angles. But to avoid all ambiguity, we shall henceforth limit our reasoning to

1ว

polygons with salient angles, which might otherwise be named convex polygons. Every convex polygon is such that a straight line, drawn at pleasure, cannot meet the contour of the polygon in more than two points.

[blocks in formation]

84. The opposite sides and angles of a parallelogram are equal.

D

B

Draw the diagonal BD. The triangles ADB, DBC, have a common side BD; and since AD, BC, are parallel, they have also the angle_ADB=DBC (67.); and since A AB, CD, are parallel, the angle, ABD=BDC: hence they are equal (38.); therefore the side AB, opposite the angle ADB, is equal to the side DC, opposite the equal angle DBC; and in like manner, AD the third side, is equal to BC: hence the opposite sides of a parallelogram are equal.

Again, since, the triangles are equal, it follows that the angle A is equal to the angle C; and also that the angle ADC, composed of the two ADB, BDC, is equal to ABC, composed of the two DBC, ABD: hence the opposite angles of a parallelogram are also equal.

Cor. Two parallels AB, CD, included between two other parallels AD, BC, are equal.

[blocks in formation]

86. If the opposite sides of a quadrilateral are respectively equal, the equal sides will be parallel, and the figure will be a parallelogram.

Let ABCD be a quadrilateral (see the last figure) having its opposite sides respectively equal, viz. AB=DC, and AD =BC; then will these sides be parallel, and the figure a parallelogram.

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; therefore, the angle ADB, opposite the side AB, is equal to DBC, opposite CD; therefore (67.) the side AD is parallel to BC. For a like reason, AB is parallel to CD: therefore the quadrilateral ABCD is a parallelogram.

14

THEOREM.

87. If two opposite sides of a quadrilateral are equal and parallel, the remaining sides will also be equal and parallel, and the figure will be a parallelogram.

Draw the diagonal BD (see the last figure). Since AB is parallel to CD, the alternate angles ABD, BDC, are equal (67.); moreover, the side BD is common, and the side AB =DC; hence the triangle ABD is equal (36.) to DBC; hence the side AD is equal to BC, the angle ADB to DBC, and consequently, AD is parallel to BC; hence the figure ABCD is a parallelogram.

[blocks in formation]

88. The two diagonals of a parallelogram divide each other into equal parts, or, mutually bisect each other.

[merged small][merged small][ocr errors][merged small]

Comparing the triangles ADO, COB, we find the side AD =CB, the angle ADO=CBO (67.), and the angle DAO= OCB; hence (38.) those triangles are equal; hence AO, the side opposite the angle ADO, is equal to OC opposite OBC; hence also DO is equal to OB.

89. Scholium. In the case of the rhombus, the sides AB, BC, being equal, the triangles AOB, OBC, 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 AOB, BOC, are equal, and therefore, that the two diagonals of a rhombus cut each other at right angles.

6

BOOK II.

THE CIRCLE, AND THE MEASUREMENT OF ANGLES.

Definitions.

90. The circumference of a circle is a curve line, all the points of which are equally distant from a point within, called the centre.

The circle is the space terminated A by this curved line.*

F

E

H

91. Every straight line, CA, CE, CD, drawn from the centre to the circumference, is called a radius or semidiameter; every line which, like AB, passes through the centre, and is terminated on both sides by the circumference, is called a diameter.

From the definition of a circle, it follows that all the radii are equal; that all the diameters are equal also, and each double of the radius.

92. A portion of the circumference, such as FHG, is called an arc.

The chord or subtense of an arc is the straight line FG, which joins its two extremities.t

93. A segment is the surface, or portion of a circle, included between an arc and its chord.

94. A sector is the part of the circle included between an arc DE, and the two radii CD, CE, brawn to the extremities of the arc.

95. A straight line is said to be inscribed in a circle, when its extremities are in the circumference, as AB.

An inscribed angle is one which, like BAC, has its vertex in the circumference, B and is formed by two chords.

*Note. In common language, the circle is sometimes confounded with its circumference: but the correct expression may always be easily recurred to, if we bear in mind that the circle is a surface which has length and breadth, while the circumference is but a line.

Note. In all cases, the same chord FG belongs to two arcs, FHG, FEG, and consequently also to two segments: but the smaller one is always meant, unless the contrary is expressed.

An inscribed triangle is one which, like BAC, has its three angular points in the circumference.

And, generally, an inscribed figure is one, of which all the angles have their vertices in the circumference. The circle is said to circumscribe such a figure.

96. A secant is a line which meets the circumference in two points. AB is a se

cant.

97. A tangent is a line which has but one point in common with the circumference. CD is a tangent.

The point M is called the point of contact.
In like manner,

[blocks in formation]

two circumferences touch each other when they have but one point in common.

A polygon is circumscribed about a circle, when all its sides are tangents to the circumference (see the diagram of 277.) in the same case, the circle is said to be inscribed in the polygon.

THEOREM.

98. Every diameter divides the circle and its circumference into two equal parts.

Let AEDF be a circle, and AB a dia

meter.

A

Now, if the figure AEB be applied to AFB, their common base AB retaining its position, the curve line AEB must fall exactly on the curve line AFB, otherwise there would, in the one or the other, be points unequally distant from the centre, which is contrary to the definition of a circle.

E

[blocks in formation]

99. Every chord is less than the diameter.

For, if the radii AC, CD, (see the last figure) be drawn to the extremities of the chord AD, we shall have the straight line ADAC+CD, or AD▲ AB.

[ocr errors]

100. Cor. Hence, the greatest line which can be inscribed in a circle is equal to its diameter.

« ΠροηγούμενηΣυνέχεια »