8. The magnitude of an angle depends on the inclination which the lines that form it have to each other, and not on the length of those lines. Thus the angle DBE is greater than the angle ABC, Fig. 3. 9. When a straight line stands on another straight line, so as to incline to neither side, but makes the angles on each side equal, then each of those angles is called a right angle and the line which stands on the other is said to be perpendicular to it. Thus ADC and BDC are right angles, and the line CD is perpendicular to AB, Fig. 4. 10. An acute angle is that which is less than a right angle, as BDE, Fig. 4. 11. An obtuse angle is that which is greater than a right angle, as ADE, Fig. 4. 12. Parallel straight lines are those which are in the same plane, and which, being produced ever so far both ways, do not meet, as AB, CD, Fig. 5. 13. A figure is a space bounded by one or more lines. 14. A plane triangle is a figure bounded by three straight lines, as ABC, Fig. 6. 15. An equilateral triangle has its three sides equal to each other, as A, Fig. 7. 16. An isosceles triangle has only two of its sides equal, as B, Fig. 8. 17. A scalene triangle has three unequal sides, as ABC, Fig. 6. 18. A right angled triangle has one right angle, as ABC, Fig. 9: in which the side AC opposite to the 19. An obtuse angled triangle has one obtuse angle, as C, Fig. 10. 20. An acute angled triangle has all its angles acute, as ABC, Fig. 6. 21. Acute and obtuse angled triangles are called oblique angled triangles. 22. Any plane figure bounded by four right lines, is called a quadrilateral. 23. Any quadrilateral, whose opposite sides are parallel, is called a parallelogram, as D, Fig. 11. 24. A parallelogram, whose angles are all right angles, is called a rectangle, as E, Fig. 12. 25. A parallelogram whose sides are all equal, and angles right, is called a square as F, Fig. 13. 26. A rhomboides is a parallelogram, whose opposite sides are equal and angles oblique, as D, Fig. 11. 27. A rhombus is a parallelogram, whose sides are all equal and angles oblique, as G, Fig. 14. 28. Any quadrilateral figure that is not a parallelogram, is called a trepezium. 29. A trepezium that has two parallel sides is called a trapezoid. 30. A right line joining any two opposite angles of a quadrilateral figure, is called a diagonal. 31. That side upon which any parallelogram, or triangle is supposed to stand, is called the base; and the per called the altitude of the parallelogram, or triangle. Thus AD is the base of the parallelogram ABEC, or triangle ABC, and CD is the altitude. Fig. 15. 32. All plane figures contained under more than four sides, are called polygons; of which those having five sides, are called pentagons; those having six sides, hexagons, and so on. 33. A regular polygon is one whose angles, as well as sides, are all equal. 34. A circle is a plane figure, bounded by one curve line called the circumference or periphery, every part of which is equally distant from a certain point within the circle; and this point is called the centre, Fig. 16. 35. The radius of a circle is a straight line drawn from the centre to the circumference, as CB, Fig. 17. 36. The diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference, as AE, Fig. 17. It divides the circle into two equal parts, called semicircles. 37. A quadrant is one quarter of a circle, as ACB, Fig. 17. Note.-The fourth part of the circumference of a circle is also called a quadrant. 38. A segment of a circle is the figure contained by a right line, and the part of the circumference it cuts off: thus AEBA and AEDA are segments of the circle ABED, Fig. 16. 59. An arc of a circle is any part of the circumference; 40. Ratio is a mutual relation between two quantities of the same kind with respect to magnitude. Note. A ratio is generally expressed, either by two numbers or by two right lines. 41. When two quantities have the same ratio as two other quantities, the four quantities taken in order are called proportionals; and the last is said to be a fourth proportional to the other three. 42. When three quantities of the same kind are such that the first has to the second the same ratio which the second has to the third, the third is called a third proportional to the first and second, and the second is called á mean proportional between the first and third. GEOMETRICAL PROBLEMS. PROBLEM I. To bisect a right line, AB, Fig. 18. Open the dividers to any distance more than half the line AB, and with one foot in A, describe the arc CFD; with the same opening, and one foot in B, describe the arc CGD, meeting the first arc in C and D; from C to D draw the right line CD, cutting AB in E, which will be equally distant from A and B. PROBLEM II. At a given point A, in a right line EF, to erect a perpendicular, Fig. 19. tances AC, AD; from C and D, as centres, with any radius greater than AC or AD, describe two arcs intersecting each other in B; from A to B, draw the lineAB, which will be the perpendicular required. PROBLEM III. To raise a perpendicular on the end B of a right line AB, Fig. 20. Take any point D not in the line AB, and with the distance from D to B, describe a circle cutting AB in E; from E through D draw the right line EDC, cutting the' periphery in C, and join CB, which will be perpendicular to AB. PROBLEM IV. To let fall a perpendicular upon a given line BC, from a given point A, without it, Fig. 21. In the line BC take any point D, and with it as a centre and distance DA describe an arc AGE, cutting BC in G; with G as a centre, and distance GA, describe an arc cutting AGE in E, and from A to E draw the line AFE; then AF will be perpendicular to AB. PROBLEM V. Through a given point A to draw a right line AB, parallel to a given right line CD, Fig. 22. From the point A to any point F, in the line CD, draw the right line AF; with F as a centre and distance FA, describe the arc AE, and with the same distance and centre A describe the arc FG; make FB equal to AE, and through A and B draw the line AB, and it will be |