A DEFINITION 11. If two adjacent angles made by two straight lines at the point where they meet be equal, each of these angles is called a right angle, and the straight lines are said to be at right angles to each other. Either of two straight lines which are at right angles to each other is said to be perpendicular to the other. E D B If a straight line AE be drawn from a point A at right angles to a given straight line CD, the part AE intercepted between the point and the straight line is commonly called the perpendicular from the point A on the straight line CD. Euclid uses as a postulate, POSTULATE 5. All right angles are equal to one another. It is not necessary to assume this proposition, since it can be proved by the method of superposition. A proof will be found on a subsequent page. (p. 37) DEFINITION 12. An angle less than a right angle is called an acute angle. An angle greater than a right angle and less than two right angles is called an obtuse angle. DEFINITION 13. A line, which is such that it can be described by a moving point starting from any point of the line and returning to it again, is called a closed line. A figure composed wholly of straight lines is called a rectilineal figure. The straight lines, which form a closed rectilineal figure, are called the sides of the figure. The sum of the lengths of the sides of any figure is called the perimeter of the figure. The point, where two adjacent sides meet, is called a vertex or an angular point of the figure. The angle formed by two adjacent sides is called an angle of the figure. A straight line joining any two vertices of a closed rectilineal figure, which are not extremities of the same side, is called a diagonal*. The surface contained within a closed figure is called the area of the figure. A closed rectilineal figure, which is such that the whole figure lies on one side of each of the sides of the figure, is called a convex figure. A closed rectilineal figure is in general denoted by naming the letters, which denote its vertices, in order: for instance the five-sided figure in the diagram is denoted by the letters A, B, C, D, E, in order: i.e. it might be called the figure ABCDE, or the figure CBAED. A, B, C, D, E are its vertices. AB, BC, CD, DE, EA are its sides. ABC, BCD, CDE, DEA, EAB are its angles. AC, AD are two of its diagonals. E It will be observed that a closed figure has the same number of angles as it has sides. If a closed figure have an even number of sides, we speak of a pair of sides as being opposite, and of a pair of angles as being opposite. If a closed figure have an odd number of sides, we speak of an angle as being opposite to a side and vice versa. For instance in the quadrilateral ABCD the side AD is said to be opposite to the side BC, and the angle BAD opposite to the angle BCD, but in the five-sided figure ABCDE the side CD is said to be opposite to the angle BAE, and the angle AED opposite to the side BC. * Derived from diá "through", and ywvía "an angle". DEFINITION 14. A figure, all the sides of which are equal, is called equilateral. A figure, all the angles of which are equal, is called equiangular. A figure, which is both equilateral and equiangular, is called regular. DEFINITION 15. A closed rectilineal figure, which has three sides*, is called a triangle. A closed rectilineal figure, which has four sides, is called a quadrilateral. A closed rectilineal figure, which has more than four sides, is called a polygon +. DEFINITION 16. A triangle, which has two sides equal, is called isosceles‡. A triangle, which has a right angle, is called right-angled. The side opposite to the right angle is called the hypotenuse §. * A figure, which has three sides, must also have three angles. It is for this reason called a triangle. + Derived from Toλús "much" and ywvía "an angle". ‡ Derived from toos “equal” and σkéλos “a leg”. § Derived from Tó "under" and Teivew "to stretch". ʼn vπoτείνουσα γραμμή right angle). the line subtending" or "stretching across (the A triangle, which has an obtuse angle, is called obtuseangled. л A triangle, which has three acute angles, is called acuteangled. DEFINITION 17. A quadrilateral, which has four sides equal, is called a rhombus. DEFINITION 18. A quadrilateral, whose opposite sides are parallel, is called a parallelo gram. DEFINITION 19. A parallelogram, one of whose angles is a right angle, is called a rectangle. It will be proved later that each angle of a rectangle is a right angle. DEFINITION 20. A rectangle, which has two adjacent sides equal, is called a square. It will be proved later that all the sides of a square are equal. DEFINITION 21. Two figures are said to be equal in all respects, when it is possible to shift one unchanged in shape and size so as to coincide with the other. The figures ABCDE, FGHKL are equal in all respects, if it be possible to shift ABCDE so that the vertices A, B, C, D, E may coincide with the vertices F, G, H, K, L respectively: in which case the sides of the two figures must be equal, AB, BC, CD, DE, EA to FG, GH, HK, KL, LF respectively, and the angles must be equal, ABC, BCD, CDE, DEA, EAB to FGH, GHK, HKL, KLF, LFG respectively. D DEFINITION 22. A plane closed line, which is such that all straight lines drawn to it from a fixed point are equal, is called a circle. This point is called the centre of the circle. It will be proved hereafter that a circle has only one centre. B E C A straight line drawn from the centre of the circle to the circle is called a radius. A straight line drawn through the centre and terminated both ways by the circle is called a diameter. It will be proved hereafter that three points on a circle completely fix the position and magnitude of the circle: hence we generally denote a circle by mentioning three points on it; for instance the circle in the diagram might be called the circle BDE, or the circle DBC. The one assumption which we make with reference to describing a circle is contained in the following postulate: |