the intersection of the lines, or by three letters, of which that at the intersection must always occupy the middle. Thus, (Fig. 1,) the angle between BA and AC may be read simply A or BAC.

31. The magnitude of an angle has no reference to the space included between the lines, nor to their length, but solely to their inclination.

Fig. 2.

D

E

32. Where one straight line stands on another so as to make the adjacent angles equal, each of these angles is called a right angle; and the lines are said to be perpendicular to each other. Thus, (Fig. 2,) if ACD = BCD, each is a right angle, and CD is perpendicular to AB.

B

33. An angle less than a right angle is called an acute angle. Thus, BCE or ECD (Fig. 2) is an acute angle.

34. An angle greater than a right angle is called an obtuse angle. ACE (Fig. 2) is an obtuse angle.

35. The distance of a point from a straight line is the length of the perpendicular from that point to the line.

36. Parallel straight lines are those of which all points in the one are equidistant from the other.

37. A figure is an enclosed space.

38. A triangle is a figure bounded by three straight lines.

39. An equilateral triangle is one the three sides of which are equal.

40. An isosceles triangle is one of which two of the sides are equal. The third side is called the base.

41. A scalene triangle has three unequal sides.

42. A right-angled triangle has one of its angles a right angle.

43. The side opposite the right angle is called the hypothenuse, and the other sides, the legs.

44. An obtuse-angled triangle has one of its angles obtuse.

45. A quadrilateral figure is bounded by four sides.

46. A parallelogram (Fig. 3) is a quadrilateral, the opposite sides of which are parallel.

Fig. 3.

47. A rectangle (Fig. 4) is a parallelogram, the adjacent sides of which are perpendicular to each

Fig. 4.

A

D

other. Thus, ABCD is a rectangle. A rectangle is read either by naming the letters around it in their order, or by naming two of the sides adjacent to any angle. Thus, the rectangle ABCD is B read the rectangle AB.BC.

Whenever the rectangle of two lines, such as DE.EF, is spoken of, a rectangular parallelogram, the adjacent sides of which are equal to the lines DE and EF, is meant.

48. A square is a rectangle, all the sides of which are equal.

49. A rhombus is an oblique parallelogram, the sides of which are equal.

50. A rhomboid is an oblique parallelogram, the adjacent sides of which are unequal.

51. All quadrilaterals that are not parallelograms are called trapeziums.

52. A trapezoid is a trapezium, having two of its sides parallel.

53. Figures of any number of sides are called polygons, though this term is generally restricted to those having more than four sides.

54. The diagonal of a figure is a line joining any two opposite angles.

56. The altitude of a figure is the distance of the highest point from the line of the base. CE (Fig. 5) is the altitude of ABCD.

57. The diameter of a circle is a straight line through the centre, terminating in the circumference.

58. The radius of a circle is a straight line drawn from the centre to the circumference.

59. A segment of a circle is any part cut off by a straight line. Thus, ABCD is a segment.

B

Α

Fig. 6.

60. A semicircle is a segment cut off by the diameter. ABC and AEB (Fig. 7) are semicircles.

61. A quadrant is a portion of a circle included between two radii at right angles to each other. ADC and BDC (Fig. 7) are quadrants.

62. The angle in a segment is the angle contained between two straight lines drawn from any point in the arc of a segment to the extremities of that arc. Thus, ABD and ACD (Fig. 6) are angles in the segment ABCD.

63. Similar rectilineal figures have their angles equal, and the sides about the equal angles proportionals.

64. Similar segments of a circle are those which contain equal angles.

SECTION II.

GEOMETRICAL PROPERTIES AND PROBLEMS.

A.-GEOMETRICAL PROPERTIES.

65. ALL right angles are equal to each other.

66. The angles which one straight line makes with another on one side of it are together equal to two right angles. Thus, ACE and ECB (Fig. 2) are together equal to two right angles. (13.1.)

67. If a number of straight lines are drawn from a point in another straight line, all the successive angles are together equal to two right angles. Thus, ACD+DCE + ECB (Fig. 2) make two right angles.

69. Triangles which have two sides and the included angle of one respectively equal to the two sides and the included angle of the other, are equal in all respects. (4.1.)

70. Triangles which have two angles and the interjacent side of one respectively equal to two angles and the interjacent side of the other, are equal in all respects. (26.1.)

71. Triangles which have two angles of the one respectively equal to two angles of the other, and which have also the sides opposite to two equal angles equal to each other, are equal in all respects. (26.1.)

72. If a straight line cuts two parallel lines, the angles similarly situated in respect to these lines, and also those alternately situated, will be equal to each other (29.1.) Thus, (Fig. 9,) EFB=FGD, BFG=DGH, AFE = CGF, and AFG CGH, C being similarly situated; and AFE

=

=

с

FGD, and BFG FGC, being alternately situated.

73. If a straight line cuts two parallel straight lines, the two exterior angles on the same side of the cutting line, and also the two interior angles, are equal to two right