Sum of the Angles of a Polygon,

and the two triangles are equiangular with respect to each other.

68. Corollary. In a triangle, there can only be one right angle, or one obtuse angle.

69. Corollary. In a right triangle, the sum of the acute angles is equal to a right angle.

70. Corollary. An equilateral triangle, being also equiangular, has each of its angles equal to a third of two right angles, or of one right angle.

71. Corollary. In any triangle ABC, if we produce the side AC toward D, the exterior angle BCD is equal to the sum of the two opposite interior angles A

and B.

72. Theorem. The sum of all the interior angles of a polygon is equal to as many times two right angles as it has sides minus two.

Proof. Let ABCDE, &c. (fig. 37), be the given polygon.

Draw from either of the vertices, as A, the diagonals AC, AD, AE, &c.

The polygon will obviously be divided into as many triangles as it has sides minus two, and the sum of the angles of these triangles is the same as that of the angles of the polygon. But the sum of the angles of each triangle is, by 65, equal to two right angles; and, consequently, the sum of all their angles is equal to as many times two right angles as there are triangles, that is, as there are sides to the polygon minus two.

73. Corollary. The sum of the angles of a quadrilateral is equal to two right angles multiplied by 4-2;

The Diagonal of a Parallelogram bisects it.

which makes four right angles; therefore, if all the angles of a quadrilateral are equal, each of them will be a right angle, which justifies the definition of a square and rectangle of § 47.

74. Corollary. The sum of the angles of a pentagon is equal to two right angles multiplied by 5-2, which makes 6 right angles; therefore, when a pentagon is equiangular, each angle is the fifth of six right angles, or of one right angle.

75. Corollary. The sum of the angles of a hexagon is equal to 2 × (6-2), or 8 right angles; therefore, in an equiangular hexagon, each angle is the sixth of eight right angles, or of one right angle. The process may be easily extended to other polygons.

76. Scholium. If we would apply this proposition to polygons, which have any angles whose vertices are directed inward, as CDE (fig. 38), each of these angles is to be considered as greater than two right angles. But, in order to avoid confusion, we shall confine ourselves in future to those polygons, which have angles directed outwards, and which may be called convex polygons. Every convex polygon is such, that a straight line, however drawn, cannot meet the perimeter in more than two points.

77. Theorem. The diagonal of a parallelogram divides it into two equal triangles.

Proof. Let ABCD (fig. 39) be the parallelogram and AC its diagonal.

The two triangles ABC and ADC are equal, since they have the side AC common, the angle BAC ACD, by

Parallel Lines at Equal Distances throughout.

§30, on account of the parallels AB and CD, and BCA = CAD, on account of the parallels BC and AD.

78. Theorem. The opposite sides of a parallelogram are equal, and the opposite angles are equal.

Proof. For the triangles ACB and ACD (fig. 39) being equal, their sides CB and AB are respectively equal to AD and DC; and the angle ABC=ADC. In the same way it might be proved that BAD=BCD.

79. Corollary. Two parallel lines comprehended between two other parallel lines are equal.

=

80. Theorem. If, in a quadrilateral ABCD (fig. 39), the opposite sides are equal, namely, AB CD, and AD BC, the equal sides are parallel, and the figure is a parallelogram.

Proof. For the triangles ABC and ACD are equal, having their three sides respectively equal; and therefore ACB CAD, whence BC is parallel to AD, by § 31 ; and BAC=ACD, whence AB is parallel to CD.

81. Theorem. If two opposite sides BC, AD (fig. 39) of a quadrilateral are equal and parallel, the two other sides are also equal and parallel, and the figure ABCD is a parallelogram.

Proof. For the triangles ABC and ACD are equal, since they have the side AC common, the side BC=AD, and the included angle BCA= CAD, on account of the parallelism of BC and AD; and therefore AB and CD must be equal and parallel.

82. Theorem. Two parallel lines are throughout at the same distance from each other.

The Circle, Radius.

Proof. The two parallels AB and CD (fig. 40), being given, if through two points taken at pleasure we erect, upon AB, the two perpendiculars EG and FH, the straight lines EG, FH will, by § 34, be perpendicular to CD; and they are also parallel and equal to each other, by arts. 35 and 79.

83. Theorem. The two diagonals of a parallelogram mutually bisect each other.

=

Proof. For the triangles (fig. 41) ADO and BOC are equal, since the side BC AD, and the angles OCB= OAD, and OBC- ODA, on account of the parallelism of BC and AD; therefore AO- OC and BO=OD.

=

84. Corollary. In the case of the rhombus (fig. 42), the triangles AOB and AOD are equal, for the sides AB=AD, BO=DO, and AO is common; therefore the angles AOB and AOD are equal, and, as they are adjacent, each of them must, by definition, § 20, be a right angle, so that the two diagonals of a rhombus bisect each other at right angles.

CHAPTER VIII.

THE CIRCLE AND THE MEASURE OF ANGLES.

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

The circle is the space terminated by this curved line. 86. Definitions. The radius of a circle is the straight

Diameter, Inscribed Lines.

line, as AB, AC, AD (fig. 43), drawn from the centre to the circumference.

The diameter of a circle is the straight line, as BD, drawn through the centre, and terminated each way by the circumference.

87. Corollary. Hence, all the radii of a circle are equal, and all its diameters are also equal, and double of the radius.

88. Theorem. Every diameter, as BD (fig. 43), bisects the circle and its circumference.

Proof. For if the figure BCD be folded over upon the part BED, they must coincide; otherwise there would be points in the one or the other unequally distant from the centre.

89. Definition. A semicircumference is one half of the circumference, and a semicircle is one half of the circle itself.

90. Definition. An arc of a circle is any portion of its circumference, as BFE.

The chord of an arc is the straight line, as BE, which joins its extremities.

The segment of a circle, is a part of a circle comprehended between an arc and its chord, as EFB.

91. Theorem. Every chord is less than the diameter. Proof. Thus BE (fig. 43) is less than DB. For, joining AE, we have BD-BA+AE, but BE< BA +AE, therefore BE<BD.

92. Definition. A straight line is said to be inscribed in a circle, when its extremities are in the circumference of the circle.