139. Cor. 1. If the sum of two angles of a triangle is subtracted from two right angles, the remainder is equal to the third angle.

140. Cor. 2. If two triangles have two angles of the one equal to two angles of the other, the third angles are equal.

141. Cor. 3. If two right triangles have an acute angle of the one equal to an acute angle of the other, the other acute angles are equal.

142. Cor. 4. In a triangle there can be but one right angle, or one obtuse angle.

143. Cor. 5. In a right triangle the two acute angles are complements of each other.

144. Cor. 6. In an equiangular triangle, each angle is one-third of two right angles, or two-thirds of one right angle.

145. The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

146. Cor. The exterior angle of a triangle is greater than either of the opposite interior angles.

147. Two triangles are equal if a side and two adjacent angles of the one are equal respectively to a side and two adjacent angles of the other.

148. Cor. 1. Two right triangles are equal if the hypotenuse and an acute angle of the one are equal respectively to the hypotenuse and an acute angle of the other.

149. Cor. 2. Two right triangles are equal if a side and an acute angle of the one are equal respectively to a side and homologous acute angle of the other.

150. Two triangles are equal if two sides and the included angle of the one are equal respectively to two sides and the included angle of the other.

151. Cor. Two right triangles are equal if their legs are equal, each to each.

152. If two triangles have two sides of the one equal respectively to two sides of the other, but the included angle of the first triangle greater than the included angle of the second, then the third side of the first is greater than the third side of the second.

153. If two sides of a triangle are equal respectively to two sides of another, but the third side of the first triangle is greater than the third side of the second, then the angle opposite the third side of the first triangle is greater than the angle opposite the third side of the second.

154. In an isosceles triangle the angles opposite the equal sides are equal.

155. Cor. An equilateral triangle is equiangular, and each angle contains 60°.

156. If two angles of a triangle are equal, the sides opposite the equal angles are equal, and the triangle is isosceles.

157. Cor. An equiangular triangle is also equilateral.

158. If two sides of a triangle are unequal, the angles opposite are unequal, and the greater angle is opposite the greater side.

159. If two angles of a triangle are unequal, the sides opposite are unequal, and the greater side is opposite the greater angle.

160. Two triangles are equal if the three sides of the one are equal respectively to the three sides of the other.

161. Two right triangles are equal if a side and the hypotenuse of the one are equal respectively to a side and the hypotenuse of the other.

162. Every point in the bisector of an angle is equidistant from the sides of the angle.

163. Every point within an angle, and equidistant from its sides, is in the bisector of the angle.

164. Cor. The locus of a point within an angle, and equidistant from its sides, is the bisector of the angle.

178. The diagonal of a parallelogram divides the figure into two equal triangles.

179. In a parallelogram the opposite sides are equal, and the opposite angles are equal.

180. Cor. 1. Parallel lines comprehended between parallel lines are equal.

181. Cor. 2. Two parallel lines are everywhere equally distant.

182. If two sides of a quadrilateral are equal and parallel, then the other two sides are equal and parallel, and the figure is a parallelogram.

183. If the opposite sides of a quadrilateral are equal, the figure is a parallelogram.

184. The diagonals of a parallelogram bisect each other. 185. Two parallelograms, having two sides and the included angle of the one equal respectively to two sides and the included angle of the other, are equal.

186. Cor. Two rectangles having equal bases and equal altitudes are equal.

187. If three or more parallels intercept equal parts on any transversal, they intercept equal parts on every transversal.

188. Cor. 1. The line parallel to the base of a triangle, and bisecting one side, bisects the other side also.

189. Cor. 2. The line which joins the middle points of two sides of a triangle is parallel to the third side, and is equal to half the third side.

190. Cor. 3. The line which is parallel to the bases of a trapezoid and bisects one leg of the trapezoid bisects the other leg also.

191. Cor. 4. The median of a trapezoid is parallel to the bases, and is equal to half the sum of the bases.

205. The sum of the interior angles of a polygon is equal to two right angles, taken as many times less two as the figure has sides.

206. Cor. The sum of the angles of a quadrilateral equals two right angles taken (4—2) times, i.e., equals 4 right angles; and if the angles are all equal, each angle is a right angle. In general, each angle of an equiangular 2(n-2) polygon of n sides is equal to

n

right angles.

207. The exterior angles of a polygon, made by producing each of its sides in succession, are together equal to four right angles.

208. A quadrilateral which has two adjacent sides equal, and the other two sides equal, is symmetrical with respect to the diagonal joining the vertices of the angles formed by the equal sides, and the diagonals intersect at right angles.

209. If a figure is symmetrical with respect to two axes perpendicular to each other, it is symmetrical with respect to their intersection as a centre.

BOOK II.

THEOREMS.

227. The diameter of a circle is greater than any other chord; and bisects the circle and the circumference.

228. A straight line cannot intersect the circumference of a circle in more than two points.

229. In the same circle, or equal circles, equal angles at the centre intercept equal arcs; conversely, equal arcs subtend equal angles at the centre.

230. In the same circle, or equal circles, equal chords subtend equal arcs; conversely, equal arcs are subtended by equal chords.

231. In the same circle, or equal circles, if two arcs are unequal, and each is less than a semi-circumference, the greater arc is subtended by the greater chord; conversely, the greater chord subtends the greater arc.

232. The radius perpendicular to a chord bisects the chord and the arc subtended by it.

233. Cor. 1. The perpendicular erected at the middle of a chord passes through the centre of the circle.

234. Cor. 2. The perpendicular erected at the middle of a chord bisects the arcs of the chord.

235. Cor. 3. The locus of the middle points of a system of parallel chords is the diameter perpendicular to them. 236. In the same circle, or equal circles, equal chords are equally distant from the centre; and conversely.

237. In the same circle, or equal circles, if two chords are unequal, they are unequally distant from the centre, and the greater is at the less distance.

238. In the same circle, or equal circles, if two chords