FORM STUDY AND MEASUREMENT ANGLES Instruments: Rule, protractor and compasses 1. Lay the compasses down flat, as shown in the picture. The two arms form an angle. Move one arm of the compasses from one position to another. How is the size of the angle changed? Does the change depend upon the length of the arms of the compasses? 2. Draw two lines forming an angle. 3. Draw three lines forming two angles arranged as in Figure 1. If the angles are 20° and 25°, what is their sum? 4. Draw two other angles; measure them with a protractor and find their sum. B Fig. 2. B A Fig. 1. = 5. Two angles, as A and B of Figure 2, are 65° and 50°. The equation A - B C means that C is the difference between A and B. How many degrees in C? 6. Draw a figure like that shown, but larger. Measure the angles A and B. Find C. Verify by measurement. Draw the following angles with the protractor, then construct the sum of each pair; the difference between each pair: Α 1. Draw a straight line 24 in. long. Place the points of the compasses at its end points. Transfer the points of the compasses to another position. How far apart are the new points? 2. Place the points of the compasses against the edge of a rule, and by means of the spaces spread the points 3 in. apart. Transfer the points of the compasses to paper. Connect the points. What is the length of the line? 22 3. Transfer these lines from the book to your paper: 4. Draw a line of convenient length, and using the length of the line as radius, draw an arc with each end as center. Connect the point of cutting with the ends of the line. What kind of a figure is formed? How do its sides compare? 5. A lot is in the shape of an equilateral triangle of side 300 ft. Draw a plan of the lot, taking 1 in. to represent 100 ft. 6. Draw a line 3 in. long. About each end as a center draw a circle of radius 5 in. Connect an intersection of these circles with the ends of the first line. B 7. How many sides of the triangle of Exercise 6 are equal? It is an isosceles triangle. 8. State a method of drawing an isosceles triangle with sides of given length. 9. Draw a line 4 in. long correspond ing to AB in the figure; by drawing arcs, find a point that is 3 in. from B and five in from A. 126 EQUALITY OF TRIANGLES 1. Draw a straight line 4 in. long. Find points that are at once 3 in. from one end and 21 in. from the other. 2. Draw a triangle whose sides are 4 in., 7 in., and 8 in. 3. Tell how to draw any triangle, given its three sides. Construct with compasses triangles having the following sides: 6. Draw two triangles, the sides of each being 3 in., 5 in., and 6 in. Cut out the triangles and apply one to the other. Can they be made to coincide exactly? 7. If three sides of a triangle are respectively equal to three sides of another, how do the triangles compare? 8. Draw two triangles with parts equal as shown in the picture. Cut them out. Can they be made to coincide? 9. If the two sides and in cluded angle of a triangle are respectively equal to the two sides and the included angle of another triangle, how do they compare in size and shape? 10. To measure the length, AB, a surveyor measured A C and C B. He laid off a distance equal to AC (Where?), and one Aequal to BC (Where?), and measured the line, A'B'. Why was it equal to A B? 11. Draw two triangles with the parts equal that are shown in the figure. Cut them out and apply one to the other. 12. If two triangles have B C ہتا two angles and the included side of one equal to two angles and the included side of the other, how do they compare? Of what angles is each of the following the difference? 10. If convenient, test these relations by drawing a figure and measuring the angles. 11. Draw two parallel lines each 4 in. long, and connect their ends so as to form a parallelogram shaped like the figure. 2 3 4 12. Draw the diagonals, or lines joining the opposite corners and number the triangles in order. 13. Cut out the figure just mentioned, and cut along the diagonals. Select the equal triangles. Mark the equal angles and equal lines. 14. Place the pieces together again to form the parallelogram. How does the intersection of the diagonals divide each one? Point out sets of equal angles in the figure. Find other equal angles by addition. 15. In any piece of paper, preferably irregular, as shown in the figure, fold a straight line, A B. Bisect AB by folding A to B. This makes the crease CD. Angle BDA is a straight angle, or an angle of 180°. 16. Compare angle BDC and many degrees are there in each? How CDA by folding. right angle (90°) with each other are said to be perpendicular. 17. How is line CD related to the line A B? 128 THE ISOSCELES TRIANGLE 1. Fold a line, A B, and the line perpendicular to it at its middle point, as CD in the figure. C 2. Fold or draw lines from C to both A and B. Compare the length of C A with C B. 3. Repeat for other points on CD. 4. What sort of triangle is ABC? Why? 5. When the figure is folded along CD, how are triangles DBC and DAC placed? How do they compare? 6. How do the angles opposite the equal sides of an isosceles triangle compare in size? 7. How does the line CD divide the angle A C B? 8. How does the altitude of an isosceles triangle divide the base? The vertical angle? The triangle? 9. Answer Exercise 8 for an equilateral triangle. 10. It has been shown (Book II, p. 216) that the sum of the 3 angles of a triangle is 180°; when one angle is given how may the sum of the other two be found? II. If the angle between the equal sides of an isosceles triangle is given, how may the other angles be found? 12. How many degrees in each angle a in Fig. 1? 13. Draw an isosceles triangle with base 3 in. and base angles 45°. How many degrees in the angle at the vertex? 14. Draw an isosceles triangle with a base of 2 in. and an angle at the vertex of 40°. G Fig. 3. 45° D 15. Find the number of de Fig. 1. A Fig. 2. grees in the angles x, T, and T in Fig. 2. 16. Find the number of degrees in the angles A, C, and B, the parts of Fig. 3 being equal isosceles triangles. |