Ex. 50. (i) Make AOB = 65° ; produce AO to C; measure 4 BOC; what is the sum of 4s AOB, BOC? Compare the results of (i), (ii), (iii); how many right angles are there in each sum? Ex. 51. If, in fig. 23, ▲ AOB = 57°, what is ▲ BOC? Check by drawing and measuring. Ex. 52. (i) If, in fig. 23, ▲ BOC = 137°, what is AOB? Ex. 53. Draw a straight line OB; on opposite sides of OB make the two angles AOB = 42°, BOC = 129°. What is their sum? Is AOC a straight line? Ex. 55. What connection must there be between the two angles in the last Ex. in order that AOC may be straight? Ex. 56. Make an AOB to C; make COD = 36°; = 36°; produce AO calculate BOC; is BOD a straight line in your figure? Give a reason. Ex. 57. From a point O in a straight line AB, draw two lines OC, OD (see fig. 25); measure the three angles; what is their sum? Ex. 58. Repeat Ex. 57, with AOB drawn in a different direction. L Ex. 59. Draw fig. 26 making ▲ BOC = 67° and ▲ B'O'D' = 29°. What is the sum of the four angles? Ex. 60. Draw fig. 27 making ▲ BOC = 67° and ▲ BOD = 29°. What is the sum of the four angles at O? Give a reason. Ex. 61. From a point O in a straight line AB, draw straight lines OC, OD, OE, OF, OG as in fig. 28. Measure the angles AOC, COD, &c. What B is their sum? Ex. 62. From a point O, draw a set of straight lines as in fig. 29, measure the angles so formed. What is their sum? How many right angles is the sum equal to ? Ex. 63. From a point O, draw a set of straight lines as in fig. 29. Guess the size of the angles so formed; Ex. 64. Draw two straight lines as in fig. 30; measure all the angles. Ex. 65. Make LAOB = 47°; produce AO to C and BO to D; measure all the angles. Ex. 66. Repeat Ex. 65 with ▲ AOB = 166°. D B fig. 30. Ex. 67. In fig. 30, if ▲ AOB = 73°, what are the remaining angles? Verify by drawing. Ex. 68. (i) In fig. 30, if AOD = 132°, what are the remaining angles? (ii) In fig. 30, if COD = 58°, what are the remaining angles ? (iii) In fig. 30, if BOC = 97°, what are the remaining angles? REGULAR POLYGONS. A Ex. 69. Describe a circle of radius 5 cm.; at its centre O draw two lines at right angles to cut the circle at A, B, C, D. Join AB, BC, CD, DA. Measure each of these lines and each of the angles ABC, BCD, CDA, DAB. A square has all its sides equal and all its angles right angles. B Ex. 70. Describe a circle of radius 5 cm.; make a set of angles each equal to 60° (i.e. (i.e. 360°); join the points where the arms cut the circle; the figure you obtain is a hexagon (6-gon), and it is said to be inscribed in the circle. Measure each angle and side. G. S. C fig. 31. at its centre fig. 32. A figure bounded by equal straight lines, which has all its angles equal, is called a regular polygon. A figure of 3 sides is called a triangle (A). The corners of a triangle or polygon are called its vertices. The perimeter of a figure is the sum of its sides. Ex. 71. What is the perimeter of a regular 6-gon, each of whose sides is 2.7 in. long? Ex. 72. In a circle of radius 5 cm, make a regular pentagon (5-gon) as in Ex. 70; the angles you make at the centre must all be equal and there will be five of them; what is each angle? Ex. 73. Calculate the angle at the centre for each of the following regular polygons; inscribe each in a circle of radius 5 cm. (i) 8-gon, (ii) 9-gon, (iii) triangle, (iv) 10-gon, (v) 16-gon. Ex. 74. Make a table of the results of Ex. 73. Ex. 75. Explain in your own words a simple construction for a regular hexagon depending on the fact you discovered in Ex. 70, that each side of the hexagon was equal to the radius of the circle. PATTERN DRAWING. Ex. 76. Copy fig. 33, taking 5 cm. for the radius of the large circle. The dotted lines are at right angles to one another. How will you find the centres of the small circles? If you describe only part of a circle, the curve you make is called an arc of the circle. Ex. 77. Copy fig. 34, taking 5 cm. for the radius of the circle. The six points on the circle are the vertices of a regular hexagon (see Ex. 75); each of these points is the centre of one of the arcs. Ex. 78. Copy fig. 35, taking 5 cm. for the radius of the circle. The centres of the arcs are the midpoints of the sides of a square inscribed in the circle. fig. 33. fig. 34. Ex. 79. Copy fig. 36, taking 5 cm. for the radius of the circle. The angles between the dotted lines are equal; what size is each of these angles? The centres of the arcs are the midpoints of the dotted lines. fig. 35. fig. 36. |