Ex. 166. PARALLELS AND PERPENDICULARS. Give instances of parallel straight lines (e.g. the flooring boards of a room, the edges of your paper). Ex. 167. Draw with your ruler two straight lines as nearly parallel as you can judge; draw a straight line cutting them as in fig. 54; measure the angles marked. These are called corresponding angles. Are they equal? fig. 54. Ex. 168. Repeat Ex. 167 two or three times drawing the cutting line in different directions. Ex. 169. Draw two straight lines which are not parallel and proceed as in Ex. 167. Are the angles equal? Ex. 170. Draw a straight line AB (see fig. 55). In AB take a point C; through C draw CD making ▲ BCD = 90° (use your set square); through A draw AE making ▲ BAE = 90°. Are AE and CD parallel? A E C D B fig. 55. Ex. 171. In the figure you obtained in the last Ex. draw two more straight lines at right angles to CD; measure the part of each of these three straight lines cut off between AE and CD; are these parts equal? Would these three parts be equal if the lines all made different angles with CD? = Ex. 172. Repeat Ex. 170 with BCD = L BAE 60° (use your set square); draw three straight lines at right angles to CD ; measure the parts cut off between AE and CD. Ex. 173. Repeat Ex. 170 with BCD =LBAE 30° (use your set square); measure as in Ex. 172. In the course of Ex. 166-173, you should have obscrved the following properties of parallel straight lines: (i) they do not meet however far they are produced in either direction. (ii) if a straight line cuts them, corresponding angles are equal. (iii) parallel straight lines are everywhere equidistant. To draw a parallel to a given line QR through a given point P by means of a set square and a straight edge. It is important that the straight edge should not be bevelled (if it is bevelled the set square will slip over it); in the figures below a ruler with an unbevelled edge is represented, but the base of the protractor or the edge of another set square will do equally well. P Q o (i) R Place a set square so that one of its edges lies along the given line QR (as at (i)); hold it in that position and place the straight (unbevelled) edge in contact with it; now hold the straight edge firmly and slide the set square along it. The edge which originally lay along QR will always be parallel to QR. Slide the set square till this edge passes through P (as at (ii)), hold it firmly and rule the line. fig. 56. This method of drawing parallels suggests an explanation of the term corresponding angles. Ex. 174. Draw a straight line QR and mark a point P; through P draw a parallel to QR. Ex. 175. Repeat Ex. 174 several times using the different edges of the set square. (See fig. 57, and Ex. 170.) Ex. 176. Near the middle of your paper draw an equilateral triangle with its sides 1 in. long; through each vertex draw a line parallel to the opposite side. If the angle between two straight lines is a right angle the straight lines are said to be at right angles to one another or perpendicular to one another. To draw through a given point P a straight line perpendicular to a given straight line QR. The difficulty of drawing a line right to the corner of a set square can be overcome as follows: KP Q R (i) O Place a set square so that one of the edges containing the right angle lies along the given line QR (as at (i)); place the straight edge in contact with the side opposite the right angle; now hold the straight edge firmly and slide the set square along it; the edge which lay along QR will always be fig. 57. parallel to QR and the other edge containing the right angle will Ex. 177. Through a given point in a straight line draw a perpendicular to that line. Ex. 178. Draw an acute-angled triangle; from each vertex draw a perpendicular to the opposite side. Ex. 179. Repeat Ex. 178 with an obtuse-angled triangle. (You will find it necessary to produce two of the sides.) Ex. 180. Describe a circle, take any two points A, B upon it, join AB; from the centre draw a perpendicular to AB; measure the two parts of AB. Ex. 181. point of each side. Draw an acute-angled triangle; from the middle side draw a straight line at right angles to that Ex. 182. Repeat Ex. 181 with an obtuse-angled triangle. PARALLELOGRAM, RECTANGLE, SQUARE, RHOMBUS. Ex. 183. Make an angle ABC = 65°, cut off BA = 2.2 in., BC= 1.8 in.; through A draw AD parallel to BC, through C draw CD parallel to BA. A four-sided figure with its opposite sides parallel is called a parallelogram. Ex. 184. Make a parallelogram two of whose adjacent sides (i.e. sides next to one another) are 6·3 cm. and 5·1 cm., the angle between them being 34°. Measure the other sides and angles. Ex. 185. Repeat Ex. 184 with the following measurements: 10.4 cm., 2.6 cm., 116°. Ex. 186. Repeat Ex. 184 with the following measurements: 10.4 cm., 2.6 cm., 64°. Ex. 187. Draw a parallelogram two of whose sides are 3.7 in., and 0-8 in., and one of whose angles is 168°. Are its opposite sides and angles equal? It will be proved later on that the opposite sides and angles of a parallelogram are always equal. Ex. 188. Construct a quadrilateral ABCD having AB = CD = 4.7 cm., AD = BC = 7.2 cm., and ▲ A = 85°. Is it a parallelogram? Ex. 189. Make a parallelogram of strips of cardboard, one pair of sides being 5 in. long and the other pair 3 in. you Ex. 190. Open one of the acute angles of the framework have just made until it is a right angle; examine the other angles. A parallelogram which has one of its angles a right angle is called a rectangle. Ex. 191. Draw a rectangle having sides =7-3 cm. and 3-7 cm. Measure all its angles. = Ex. 192. Draw a parallelogram having sides 9-2 cm. and 4.3 cm., and one angle = 125°. Draw its diagonals, and measure their parts. Ex. 193. Repeat the last Ex. with the following measurements, 8.6 cm., 6·8 cm., 68°; test any facts you noted in that Ex. Ex. 194. Draw a parallelogram and measure the angles between its diagonals; are any of them equal? Give a reason. Ex. 195. Draw a rectangle having sides = 3.5 in. and 2-3 in. Measure its diagonals. Ex. 196. Repeat the last Ex. with the following measurements, (i) 8.6 cm., 11·2 cm., (ii) 14·3 cm., 2.8 cm. A rectangle which has two adjacent sides equal is called a square. Ex. 197. Draw a square having one side all its sides and angles. Ex. 198. Draw a square having each side its diagonals and the angles between them. 5.6 cm. Measure 3.2 in. Measure Ex. 199. Explain how you would test by folding whether a pocket handkerchief is square. Ex. 200. Make a paper square by folding. A parallelogram which has two adjacent sides equal is called a rhombus. |