Hence, when one line stands on another, the two adjacent angles are supplements of each other. Hence a right angle is equal to its supplement. The supplement of an acute angle is obtuse, and, conversely, the supplement of an obtuse angle is acute. 26. A reflex angle is an angle which is greater than a straight angle, and less than two straight angles, as the angle 0. Acute, obtuse, and reflex angles are called oblique angles, in distinction from right and straight angles; and Fig. 13 intersecting lines which are not perpendicular to each other are called oblique lines. 27. Where two angles are contained between two intersecting lines on opposite sides of the A vertex, they are called opposite or vertical angles. Thus, AOC and BOD are opposite or vertical angles, as also AOD and COB. B Fig. 14 ANGULAR MEASURE. 28. A right line drawn from the vertex and turning about it in the plane of the angle from the position of coincidence with one side of the angle to that of coincidence with the other side, is said to turn through the angle, and. the angle is the greater as the quantity of turning is greater. Thus, suppose that the right line OP (Fig. 15) is capable of revolv B ing about the point 0, like the E hands of a watch, but in the opposite direction, and that it has A A passed successively from the position 0 A to the positions occupied by OB, OC, OE, etc. Then it is clear that the line must have done Fig. 15 more turning in passing from OA to OC than in passing F from OA to OB; and consequently the angle AOC is said to be greater than the angle AOB. When the revolving line OP turns from coincidence with OA to the position OB it is said to describe or to generate the angle AOB. When the revolving line has turned from the position 0A to the position OD, perpendicular to OA, it has generated the right angle AOD; when it has turned to the position OA', it has generated the straight angle AOA'; when it has turned to the position OF, it has generated the reflex angle AOF; when it has turned entirely around to the position OA, it has generated two straight angles. Hence, the whole angle which a line must turn through, about a point in a plane, to take it around to its first position, is two straight angles, or four right angles. Again, since the revolving line may turn from one position to the other in either of two B directions, two angles are formed by. two lines drawn from a point. A Thus, if 0A, OB be the sides of an angle, a line may turn from the position OA to the position OB about the point 0 in either of the two directions indicated by the arrows, giving the obtuse angle AOB (marked a), and the reflex angle AOB (marked b). Angles are generally measured in degrees, minutes, and seconds. A degree is the ninetieth part of a right angle, or the three hundred and sixtieth part of four right angles. A minute is the sixtieth part of a degree; and a second is the sixtieth part of a minute. Degrees, minutes, and seconds are denoted by the symbols , ', ". Thus 7 degrees, 24 minutes, and 38 seconds is written, 7° 24' 38". Hence, when the revolving line OP (Fig. 15) has turned through one-fourth of a revolution, it has generated a right angle, or 90°; when it has made half a revolution, it has generated a straight angle, or 180°; and when it has made a Fig. 16 a whole revolution, it has generated two straight angles, or 360° SUPERPOSITION. 29. The placing of one geometric magnitude on another, such as a line on a line, an angle on an angle, etc., is called superposition. The superposition employed in Geometry is only mental, that is, we conceive one magnitude to be taken up and laid down upon the other; and then, if we can prove that they coincide, we infer that they are equal. This is the ultimate test of the equality of two geometric magnitudes. Thus, if two straight lines are to be compared, we conceive one to be taken up and placed on the other, and find whether their two ends can be made to coincide. If so, they are equal; if not, they are unequal. If two angles are to be compared, we conceive one to be taken up and applied to the other. If they can be so placed that their vertices coincide in position and their sides in direction, the angles are equal (18). Superposition involves the principle* that “any figure may be taken up, transferred from one position to another, and laid down again without change of form or size." Magnitudes which coincide with one another throughout their whole extent are said to be equal. DEFINITIONS OF TERMS. 30. A theorem is a truth which requires proof. 31. A problem is a question which requires solution, such as a particular line to be drawn, or a required figure to be constructed. 32. An axiom is a self-evident truth, which is admitted without proof. 33. A postulate assumes the possibility of solving a certain problem. * Euclid makes frequent use of this principle, without explicitly stating it.--CASEY. 34. A proposition is a general term for a theorem, problem, axiom, or postulate. 35. A demonstration is the course of reasoning by which we prove a theorem to be true. 36. A corollary is a conclusion which follows immediately from a theorem. 37. A lemma is an auxiliary theorem required in the demonstration of a principal theorem. 38. A scholium is a remark upon one or more propositions. 39. An hypothesis is a supposition made either in the enunciation of a proposition or in the course of a demonstration. 40. A solution of a problem is the method of construction which accomplishes the required end. 41. A construction is the drawing of such lines and curves as may be required to prove the truth of a theorem, or to solve a problem. 42. The enunciation of a theorem consists of two parts : the hypothesis, or that which is assumed; and the conclusion, or that which is asserted to follow therefrom. Thus, in the typical theorem, If A is B, then C is D, the hypothesis is that A is B, and the conclusion that C is D. 43. The enunciation of a problem consists of two parts: the data, or things supposed to be given; and the quæsita, or things required to be done. 44. Two theorems are said to be converse, each of the other, when the hypothesis of each is the conclusion of the other. Thus, If C is D, then A is B, is the converse of the typical theorem in (42). POSTULATES. 45. Let it be granted 1. That a straight line may be drawn from any one point to any other point. 2. That a terminated straight line may be produced to any distance in a straight line. 3. That a circle may be described with any centre, at any distance from that centre. AXIOMS. 46. 1. Things which are equal to the same thing are equal to each other. 2. If equals be added to equals the sums will be equal. 3. If equals be taken from equals the remainders will be equal. 4. If equals be added to unequals the sums will be unequal. 5. If equals be taken from unequals the remainders will be unequal. 6. Things which are double the same thing, or equal things, are equal to one another. 17. Things which are halves of the same thing, or of equal things, are equal to one another. 8. The whole is greater than any of its parts. GEOMETRIC AXIOMS.. 10. A straight line is the shortest distance between any two points. 11. If two straight lines have two points in common, they will coincide throughout their whole length, and form but one straight line. 12. Through a given point only one straight line can be drawn parallel to a given straight line. |