45. Problem. Construct an angle of given size at a point in Place the protractor with its center on P and its edge on PB as in the figure. Then place a point R on the paper opposite the 35° mark on the protractor. Remove the protractor and draw the line PR. BPR equals 35°. Ex. 81. Construct with the protractor an angle of : (a) 70°; (b) 40°; (c) 65°; (d) 100°; (e) 143°. Then angle Write below each angle whether it is an acute or an obtuse angle. 46. A Field Protractor. The figure at top of page 20 represents a simple field protractor which can be made by some member of the class. With it angles can be measured in the field and thus some elementary surveying problems can be solved. On a flat board about 20 inches square, draw a circle of diameter 10 inches. Divide its circumference into 360 equal parts. Make an arm which may swing about the center of the circle as pivot. Let the arm have upon it two "sights" directly in line with the center of the circle. At the end of the arm and in line with the sights place a pin. The board may be attached to the end of a stake about 4 feet long, or better to a tripod. This instrument can be used to measure angles in the open field. Thus, to measure an strument over point A. ▲ CAB, place the inMake the board stand level. (An inexpensive level would be a great help.) Holding the board stationary, sight first at point C, and read the angle on the protractor; then sight at point B, and note the angle on the protractor. The number of degrees through which the arm is turned in passing from AC to AB is the measure of angle CAB. 47. Triangle (A). Three points which do not lie in the same straight line determine three segments. Thus, A, B, and C determine the segments AB, BC, and AC. The figure formed by these three segments is called the triangle ABC (ABC). A, B, and C are the vertices of the triangle; AB, BC, and AC are the sides of the triangle; ≤ A, ▲ B, and ≤ C are the angles of the triangle. B The sides and angles of the triangle are called the parts of the triangle. They are six in number. Opposite each side there is an angle, and opposite each angle there is a side. Thus, C is opposite side AB. 48. Experimental Geometry. Many facts about geometrical figures can be discovered by careful drawing, measurement, and observation. Ex. 82. On a line AB, at a point P, draw a ray PC making ▲ APC = 80°. Measure CPB. What fact studied previously does this exer cise verify? Ex. 83. Draw two intersecting straight lines. Measure each of a pair of vertical angles. What fact studied previously does this exercise verify? Ex. 84. long and Construct a ▲ ABC, having AB and BC each 3 inches B = 40°. Measure A and C. How do they compare? (A triangle having two equal sides is called an isosceles triangle.) Ex. 85. Draw any triangle having two equal sides. angles opposite the equal sides. How do they compare? suggested by Exercises 84 and 85 ? Ex. 86. Draw any triangle of reasonably large size. of its angles. Find their sum. Measure the What fact is Measure each Repeat the exercise for another triangle Compare your results with those of some What seems to be the sum of the angles of a triangle? of somewhat different shape. other pupils. Ex. 87. Draw a ▲ ABC in which AB and BC are each 3 inches and LB = 50°. Let E be the mid-point of BC, and F the mid-point of AB. Draw AE and CF. Measure them. What seems to be true? Ex. 88. Draw any triangle ABC of reasonably large size. Let E be the mid-point of AB, and F the mid-point of BC. Draw EF. Compare EF and AC by measuring them. What seems to be the relation between them? Ex. 89. Draw a segment AB. At its center, C, draw a line CD perpendicular to AB. From E, any point on CD, draw AE and BE. Compare them by means of your dividers. Take any other point on CD and repeat the exercise. What fact seems to be suggested? Ex. 90. Let AB be any line segment. Draw CALAB at A, and DB 1 AB at B. Make CA= DB. Draw AD and CB. Compare them either by measurement or by means of the dividers. 49. Objections to studying geometry only by the experimental method may be given. To get satisfactory results, the figures must be drawn and measured with greater accuracy than is usually possible. Conclusions reached from the study of one or two special figures may be incorrect. Frequently one is misled by assuming relations which appear to the eye to be correct. Ex. 91. In the first figure above, are AB and CD straight lines? 50. Demonstrative Geometry. For the reasons given in § 49 and for other reasons, it is customary to study geometry by what is known as the demonstrative method. Statements are not accepted until they are proved to be true, except for a few fundamental ones which are assumed as a foundation. 51. An Axiom is a statement accepted as true without proof. Usually the truth is very evident. The following are important axioms; others will be introduced as they become necessary. Ax. 1. AXIOMS Quantities which are equal to the same quantity or to equal quantities are equal to each other. (See Ex. 22 and Ex. 23.) Ax. 2. Any quantity may be substituted for its equal in a mathematical expression. Ax. 3. If equals be added to equals, the sums are equal. (See Ex. 24.) Ax. 4. If equals be subtracted from equals, the remainders are equal. Ax. 5. If equals be multiplied by equals, the products are equal. Ax. 6. If equals be divided by equals, the quotients are equal. (The divisor must not be zero.) Ax. 7. The whole equals the sum of its parts. Ax. 8. The whole is greater than any of its parts. Ax. 9. If a and b are any two magnitudes of the same kind, then a is less than b, is equal to b, or is greater than b. Ax. 10. Only one straight line can be drawn through two points. (§ 8.) Ax. 11. The straight line segment is the shortest line that can be drawn between two points. (§ 14.) Ax. 12. All right angles are equal. (§ 27.) Ax. 13. An angle has only one bisector. (§ 23.) 52. A Theorem is a statement which requires proof. Every theorem can be expressed by a sentence which has one clause beginning with "if" and a second clause beginning with "then." The clause beginning with "if" is called the Hypothesis; it indicates what is known or assumed. The clause beginning with "then" is called the Conclusion; it states what is to be proved. Thus (Hypothesis) If two sides of a triangle are equal, (Conclusion) then the angles opposite are equal. 53. Some theorems have been proved already in an informal manner. INFORMALLY PROVED THEOREMS 1. Two straight lines can intersect at only one point. (§ 11.) 2. All radii of the same circle or of equal circles are equal. (§ 17.) 3. A straight angle equals two right angles. (§ 32.) 4. All straight angles are equal. (§ 33.) 5. The sum of all the successive adjacent angles around a point on one side of a straight line is one straight angle. (§ 34.) 6. The sum of all the successive adjacent angles around a point is two straight angles. (§ 35.) 7. Complements of the same angle or of equal angles are equal. (§ 37.) 8. If two adjacent angles have their exterior sides in a straight line, they are supplementary. (§ 39.) 9. If two adjacent angles are supplementary, their exterior sides lie in a straight line. (§ 40.) 10. Supplements of the same angle or of equal angles are equal. (§ 41.) 54. In a formal demonstration or proof, each statement made is proved by quoting a definition, an axiom, the hypothesis, or some previously proved theorem. |