For this purpose a protractor is used. The simplest form is the semicircular protractor shown in the illustration. To measure the angle place the point Cat D, the vertex of the angle, and the point B on the line ED. The number of degrees on the protractor at the point cut by the line DG gives the measure of the angle. The protractor has two sets of figures each extending to 180°. In measuring the angle GDE, the point B of the protractor D 50 G E being on DE, use the lower row of figures which extends from 0° at B' to 180° at A'. Angles Construction Exercises 1. Draw two lines in such a way as to make (a) four angles. (b) Two angles. (c) One angle. 2. Using the protractor, draw two lines making (a) an angle of 40°. (b) An angle of 75°, one line being vertical. (c) Two lines making an angle of 120°, both lines being oblique. 3. (a) Draw two lines forming two angles, one of which contains 50°, and write in each angle the number of degrees it contains. (b) Draw two lines forming two angles, one of which measures 110°, and write in each angle the number of degrees it contains. (c) What is the sum of the two supplementary angles in each case? 4. Draw (a) two lines making two equal supplementary angles. (b) Two lines forming four equal angles. (c) Mark in each of the six angles the number of degrees it contains. NOTE.-A line is perpendicular to another when the supplementary angles are equal. 5. Using the protractor, draw a perpendicular (a) to a horizontal line. (b) To a vertical line. (c) To an oblique line. 6. (a) Draw two lines cutting each other at an angle of 60°, and write in each of the four angles the number of degrees it contains. (b) Draw two lines cutting each other at an angle of 140°, and write in each of the four angles the number of degrees it contains. 7. (a) To a horizontal line draw two lines an inch apart, each making an angle of 90° with the first line. (b) Where will the last two lines intersect each other? (c) Using the protractor, draw two lines perpendicular to a vertical line. (d) Two perpendicular to an oblique line. 8. By means of the ruler and the triangle draw (a) several perpendiculars to a given line. (b) Several oblique lines parallel to each other. 40° 9. To a given line draw a second line making an angle of 40° with the first. By means of the ruler and the triangle draw a third line parallel to the second. (a) How many degrees are contained in the angle made by the third line with the first line on the same side? (6) How many degrees are there in each of the two supplementary angles? 10. By means of the protractor draw several lines running in the same direction and each making an angle of 60° with a vertical line. Where will the oblique lines intersect? 11. Draw two parallel lines, using the ruler and the triangle. Draw a line 50 intersecting one of the parallel lines at an angle of 50°. Write in each of the eight angles the number of degrees it contains. Triangles 12. On a line 3 inches long construct a triangle having two base angles of 60 degrees each. (a) Measure the third angle. (b) Measure each of the other two sides. (c) Write in each angle the number of /60° degrees it contains and the length of 3 in. 60 each side. (d) How many degrees are there in the three angles? Draw the base line the given length. ciently long to intercept the third line. line unnecessarily. Make the next line suffi- 13. (a) Construct a triangle having base angles of 65° each. (b) Mark in each angle the number of degrees it contains. (c) How many degrees are there in the three angles? How do the other sides compare in length? (d) With the base? (e) With each other? 14. Construct an isosceles triangle having its base (a) vertical. (b) Oblique. (c) Having its vertex below the base. 15. Construct a triangle having one base angle of 50° and one of 60°. (a) Measure the third angle and write in it the number of degrees it contains. Measure the three sides. (b) The longest side is opposite which angle? (c) The shortest side is opposite which angle? (d) How many degrees do the three angles contain? 16. (a) Construct a triangle containing one right angle. (b) Find the number of degrees contained in the two oblique angles. (c) Try to construct a triangle containing two right angles. (d) Construct a triangle containing an obtuse angle. (e) Find the number of degrees contained in the three angles. (f) Try to construct a triangle containing two obtuse angles. Oral Exercises 1. How many degrees are there in the sum of the three angles (a) of a right triangle? (b) Of an isosceles triangle? (c) Of an obtuse-angled triangle? (d) Of a scalene triangle? (e) Of an equilateral triangle? 2. How many degrees are there in each angle of an equilateral triangle? 3. How many degrees are there in an angle supplementary to one of (a) 60°? (b) 90°? (c) 140°? (d) 30°? 4. (a) If one angle of a right triangle contains 30°, how many degrees are there in the other oblique angle? (b) If one of the angles of an isosceles triangle is 120°, how many degrees are there in each of the other two angles? (c) How many degrees are there in each angle of an isosceles right triangle? (d) When one of the base angles of an isosceles triangle contains 40°, how many degrees are there in each of the other two angles? 5. Give the number of degrees in the third angle of a triangle when two of the angles contain, respectively, (a) 30° and 40°. (b) 50° and 60°. (c) 70° and 80°. (d) 30° and 50°. Quadrilaterals Construction Exercises 1. Using the protractor, draw (a) a square. (b) A rectangle. Using the triangle, draw (c) a square on a 3-inch oblique line. (d) A rectangle 3 inches by 2 inches having an oblique line for the base. 3 60 " 4" 2. (a) Draw a line 3 inches long making an angle of 60° with a 4-inch line. (6) Using the ruler and the triangle, complete the parallelogram. (c) Write in each of the four angles its contents in degrees. (d) Write on each of the four sides its length. 3. On a base 3 inches long, construct a parallelogram having a side 2 inches long and containing (a) an angle of 30°. (b) Draw a line constituting its altitude. (c) Mark the length of the altitude and of each of the two remaining sides. (d) Mark in each angle of the parallelogram the number of degrees it contains. (e) Construct a parallelogram having sides of the same length as the foregoing but containing an angle of 53°. (ƒ) What is its altitude? (g) Draw a parallelogram having sides of 2 and 3 inches, respectively, and an angle of 37°. (h) What is its altitude? () Write in each of the three foregoing parallelograms its area. 4. Construct a rhombus having sides of 3 inches and (a) an angle of 60°. (b) An angle of 150°. tude 2 inches. (c) Alti 5. Construct three trapezoids of different shapes each having an altitude of 21 inches and parallel sides 3 inches and 4 inches, respectively. 6. Construct three trapeziums of different shapes each having a diagonal of 4 inches and perpendiculars of 2 inches and 3 inches, respectively, from this diagonal to the angles opposite. 7. (a) From each extremity of a 2-inch line draw two lines, each 21 inches long and making with the first line angles of 70° and 110°, respectively. (b) Complete the quadrilateral. (e) What kind of quadrilateral is it? 8. (a) From each extremity of a 2-inch line draw two lines making angles of 80° and 100°, respectively, and measuring 21 and 3 inches, respectively. (b) Complete the quadrilateral. (c) What kind of quadrilateral is it? |