1. Magnitudes which are equal to the same magnitude, or equal magnitudes, are equal to each other. 2. If equals are added to equals, the sums are equal, 3. If equals are taken from equals, the remainders are equal. 4. If equals are added to unequals, the sums are unequal in the same order; if unequals are added to unequals in the same order, the sums are unequal in that order. 5. If equals are taken from unequals, the remainders are unequal in the same order; if unequals are taken from equals, the remainders are unequal in the reverse order. 6. The doubles of the same magnitude, or of equal magnitudes, are equal; and the doubles of unequals are unequal. 7. The halves of the same magnitude, or of equal magnitudes, are equal; and the halves of unequals are unequal. 8. The whole is greater than any of its parts. 9. The whole is equal to the sum of all its parts. Q.E.D. stands for quod erat demonstrandum, which was to be proved. REFERENCES TO PLANE GEOMETRY. 83. At a given point in a given line there can be but one perpendicular to the line. 84. The complements of the same angle or of equal angles are equal. 88. The sum of all the angles about a point in a plane is equal to a perigon, or two straight angles. 93. If one straight line intersects another straight line, the vertical angles are equal. 95. Two straight lines drawn from a point in a perpendicular to a given line, cutting off on the given line equal segments from the foot of the perpendicular, are equal and make equal angles with the perpendicular. 97. The perpendicular is the shortest line that can be drawn to a straight line from an external point. 100. The sum of two lines drawn from a point to the extremities of a straight line is greater than the sum of two other lines similarly drawn, but included by them. 103. Two parallel lines are lines that lie in the same plane and cannot meet however far they are produced. 104. Two straight lines in the same plane perpendicular to the same straight line are parallel. 105. AXIOM. Through a given point only one straight line can be drawn parallel to a given straight line. 107. If a straight line is perpendicular to one of two parallel lines, it is perpendicular to the other also. 111. When two straight lines in the same plane are cut by a transversal, if the alternate-interior angles are equal, the two straight lines are parallel. 114. When two straight lines in a plane are cut by a transversal, if the exterior-interior angles are equal, these two straight lines are parallel. 117. A triangle is a portion of a plane bounded by three straight lines. The bounding lines are called the sides of the triangle, and their sum is called the perimeter; the angles included by the sides are called the angles of the triangle, and the vertices of these angles, the vertices of the triangle. 128. The homologous sides and the homologous angles of equal triangles are equal. 138. The sum of two sides of a triangle is greater than the third side, and their difference is less than the third side. 141. 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. 142. Two right triangles are equal if a leg and an acute angle of the one are equal, respectively, to a leg and the homologous acute angle of the other. 143. 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. 144. Two right triangles are equal if their legs are equal, each to each. 145. In an isosceles triangle the angles opposite the equal sides are equal. 147. If two angles of a triangle are equal, the sides opposite the equal angles are equal, and the triangle is isosceles. 149. The perpendicular from the vertex to the base of an isosceles triangle bisects the base, and bisects the vertical angle of the triangle. 150. Two triangles are equal if the three sides of the one are equal, respectively, to the three sides of the other. 151. Two right triangles are equal if a leg and the hypotenuse of the one are equal, respectively, to a leg and the hypotenuse of the other. 155. 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. 160. The perpendicular bisector of a given line is the locus of points equidistant from the extremities of the line. 161. Two points each equidistant from the extremities of a line determine the perpendicular bisector of the line. 166. A parallelogram is a quadrilateral which has its opposite sides parallel. 176. Two angles whose sides are parallel, each to each, are either equal or supplementary. 178. The opposite sides of a parallelogram are equal. 179. A diagonal divides a parallelogram into two equal triangles. 180. Parallel lines comprehended between parallel lines are equal. 183. 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. 185. Two parallelograms 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. |