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ELEMENTS OF PLANE GEOMETRY.
Mathematics is that branch of science which treats of Measurable Quantity.
Geometry is a branch of mathematics which treats of that species of quantity called Magnitude.
Magnitudes are of one, two, or three dimensions; as lines, surfaces, and solids. They have no material existence, but they may be represented by diagrams.
That branch of Geometry which refers to magnitudes described upon a plane, is called Plane Geometry.
DEFINITIONS OF MAGNITUDES.
1. A point is that which has position, but not magnitude.
2. A line is length without breadth.
Hence the extremities of a line are points; and the intersections of lines are also points.
Lines are named by two letters placed one at each extremity. Thus, the line drawn here is named the line AB.
3. Lines which cannot coincide in two points, without coinciding altogether, are called straight lines, as AB. A part of a line is called a segment of it; thus MN, a part of the line MP, is called a segment of MP, and NP is another segment of it.
Hence two straight lines cannot enclose a space. Neither can two straight lines have a common segment; that is, they cannot coincide in part, without coinciding altogether.
Thus if the lines ABC and ABD have a
common segment, AB, they cannot both be straight lines.
4. A crooked or broken line is composed of two or more straight lines.
5. A line of which no part is a straight line, is called a curved line, a curve line, or curve.
6. A mixed line is one composed of straight and curve lines.
7. A convex or concave line is such that it cannot be cut by a straight line in more than two points; the con
cavity of the intercepted portion is turned towards
the straight line, and the convexity from it.
8. A superficies or surface has only length and breadth. The extremities of a superficies are lines; and the intersections of one superficies with another are also lines.
9. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.
10. A plane rectilineal angle is the inclination of two straight lines which meet together, but are not in the same straight line.
The lines containing an angle are called its sides, and the point at which it is formed the angular point.
Thus the lines BC and BD are the sides of the angle contained by them, and B is the angular point.
An angle is named by the three letters used for naming its sides, the letter at the angular point being placed in the middle. Thus the angle CBD or DBC is the angle contained by the lines BC and BD; and ABC or CBA, the angle contained by AB and BC.
When only one angle is formed at a point, it may be named by: only one letter, as the angle at E.
An angle may also be named by means of a small letter placed within the angle, and near to the angular point.
Thus, angle DBC may be called angle n; and ABD, angle m. 11. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it.
12. The two adjacent angles which one straight line makes with another on the same side of it are
called supplementary angles.
Thus the angles ACD and BCD, which CD makes with AB, are said to be supplementary.
The difference between an angle and a right angle is called its complement; thus, if ABC (fig. to def. 10) is a right angle, m and n are complementary angles.
13. An obtuse angle is that which is greater than a right angle, as 0.
14. An acute angle is that which is less than a right angle, as A.
15. When an angle is not a right angle, it is said to be oblique; also two angles are said to be of the same species when they are both less or both not less than a right angle.
16. A figure is that which is enclosed by one or more boundaries. The space contained within the boundary of a plane figure is called its surface; and its surface in reference to that of another figure, with which it is compared, is called its area.
17. A circle is a plane figure contained by one line, which is called the circumference, and is such, that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.
18. This point is called the centre of the B circle, as C.
Any other point within the circle is an eccentric point.
19. A line drawn from the centre to the circumference of a circle is called a radius, as CD, CB, or CE.
20. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference, as BE.
21. A semicircle is the figure contained by a diameter, and the part of the circumference cut off by the diameter.
22. Rectilineal figures are those which are contained by straight lines.
23. Trilateral figures, or triangles, are contained by three straight lines.
24. Quadrilateral figures are contained by four straight lines. 25. Multilateral figures, or polygons, are contained by more than four straight lines.
26. Of three-sided figures, an equilateral triangle is that which has three equal sides, as E.
27. An isosceles triangle is that which has only two sides equal, as I.
28. as S.
A scalene triangle is that which has three unequal sides,
29. A right-angled triangle is that which has a right angle, as R.
30. An obtuse-angled triangle is that which has an obtuse angle, as O.
31. An acute-angled triangle is that which has three acute angles, as A.
32. When two straight lines intersect each other, the opposite angles at this point are called vertical angles; thus AEC and DEB are vertical angles; and also AED and CEB.
34. Parallel straight lines are such as are in the same plane, and which, being produced ever so far both ways, do not meet.
33. When one side of a triangle is produced, the outward angle thus formed is called the exterior angle; thus ACD (see fig. Prop. xvii. Book I.) is an exterior angle of the triangle ABC.
36. The angles contained by a secant and two parallels lying between the parallels A are called interior angles, as AGH, BGH, CHG, and GHD; and the angles contained by a secant and parallel lines lying towards the outside of these lines are called exterior angles, as AGE, BGE, CHF, and DHF.
35. A straight line cutting or falling upon two parallel lines is called a secant, as EF.
37. Alternate angles are those interior angles that lie on opposite sides of the secant, as AGH and DHG, or BGH and CHG.
38. When any rectilineal figure has all its angles equal, it is said to be equiangular; and when all its sides are equal, it is said to be equilateral.
39. When the angles of one rectilineal figure are respectively equal to those of another, the figures are said to be equiangular;
and if their sides are respectively equal, they are said to be equilateral. 40. A square is a four-sided figure which has all its sides equal, and all its angles right angles, as S.
41. An oblong or rectangle is a four-sided
figure which has all its angles right angles, but has not all its
sides equal, as R.
42. A rhombus is a quadrilateral which has all its sides equal, but its angles are not right angles, as B.
43. A parallelogram is a quadrilateral, of which the opposite
sides are parallel, as P.
44. A trapezoid is a quadrilateral, hav
ing only two sides parallel, as D.
45. All other four-sided figures besides
these are called trapeziums.
46. When an angle of a rectilineal figure is less than two right angles, it is called a salient angle, as A; and when greater than two right angles, it is said to be re-entrant, as B.
47. Any side of a rectilineal figure may be called the base. In a right-angled triangle, the side opposite to the right angle is called the hypotenuse; either of the other two sides, the base; and the other, the perpendicular. In an isosceles triangle, the side which is neither of the equal sides is called the base. The angular point opposite to the base of a triangle is called the vertex; and the angle at the vertex, the vertical angle.
48. The altitude of a triangle, or a parallelogram, is the length of a perpendicular drawn from the opposite angle or side upon the base.
49. A straight line joining two of the opposite angular points of a quadrilateral is called a diagonal.
1. Let it be granted that a straight line may be drawn from any one point to any other point;
2. That a terminated straight line may be produced to any length in a straight line;
3. And that a circle may be described from any centre, and with any radius.
1. Things which are equal to the same thing are equal to one another.
2. If equals be added to equals, the wholes are equal.