sum of two adjacent angles, BAD, DAC, whose exterior sides lie in a straight line. 30. When the adjacent angles formed by one straight line meeting another are equal, each angle is called a right angle. Thus, if the angles BAC, CAD, formed by CA meeting BD in A, are equal, each of them is a right angle. 31. A right angle is equal to one half of a straight angle. This follows from the preceding definitions; for the two equal angles, CAB, CAD, make up the straight angle BAD. 32. An acute angle is less than a right angle; an obtuse angle is greater than a right angle and less than a straight angle. Thus BAC is an acute, and CAD an obtuse, angle. D A B Acute angles and obtuse angles are called oblique angles; and lines are said to be perpendicular or oblique to each other according as they meet at a right or an oblique angle. 33. If the sum of two angles is equal to a right angle, each is called the complement of the other, and the angles are said to be complementary. If the sum of two angles is equal to a straight angle, each is called the supplement of the other, and the angles are said to be supplementary. 2. If you fold a piece of note paper so as to form an edge, what sort of a line is formed? 3. If you fold the paper again, so as to double that edge upon itself, what angle will the second edge thus formed make with the first? Geom.-2 4. If you suspend a weight by a string, in what sort of a line is the string stretched? 5. If you whirl the weight round at the end of the string, in what sort of a line does the weight move? 6. What sort of a surface is presented, roughly speaking, by the walls of a room? By the surface of a floor? By the surface of a slate? Mention other like surfaces. 7. How would you apply a straightedge or ruler so as to ascertain whether a given surface is a plane? What property of planes do you apply (Art. 9) ? 8. Straight lines can be drawn on the surface of a stovepipe, and yet it is not a plane: why not? 9. Can a straight line be drawn on the surface of an eggshell ? If not, what kind of a line can be drawn on such a surface ? 10. From a point O draw lines OA, OB, OC, OD, in one plane. Name each of the angles thus formed. Which are adjacent angles? Name one that is the sum of two; of three. 11. Can you draw two angles that have a common vertex and a common side, and yet are not adjacent ? 12. What sort of an angle is less than its supplement? Is equal to its supplement? Is greater than its supplement? PROPOSITIONS. The truths of geometry are presented for consideration under the form of general statements called propositions. 34. A theorem is a proposition stating a geometrical truth. 35. A problem is a proposition stating a proposed construction. 36. A corollary is a theorem that follows so plainly as a consequence from a preceding proposition, or definition, that its formal proof is either omitted or is merely indicated. Thus in Arts. 15, 16, 17, are given certain important corollaries from the definition of the straight line; and in Art. 23 we have a very obvious corollary from the definitions of circumference and radius. 37. An axiom is a theorem assumed as self-evident. 38. A postulate is a problem assumed as possible. 39. A scholium is a remark upon a preceding proposition. 40. The axioms and postulates, together with the definitions, constitute the logical basis of geometry. Axioms express certain simple truths in regard to magnitude in general, truths so confirmed by all our experience that the mind cannot conceive their opposites as true. All the axioms except the last two, which are really definitions of the terms whole and part, might be deduced from the first. They are such obvious truths, however, that it is deemed sufficient to state them for convenience of reference. AXIOMS. 1. Magnitudes equal to the same or equal magnitudes are equal to each other. 2. If equals are added to equals, the sums are equal. 3. If equals are taken from equals, the remainders are equal. 4. If equals are added to unequals, the sums are unequal. 5. If equals are taken from unequals, or unequals from equals, the remainders are unequal. 6. The doubles of equals are equal. 7. The halves of equals are equal. 8. The whole is greater than any of its parts. 9. The whole is equal to the sum of all its parts. POSTULATES. Let it be assumed that, in a given plane, 1. A straight line can be drawn joining any two given points. 2. A given straight line can be produced to any extent. 3. On the greater of two straight lines a part can be laid off equal to the less. 4. A circumference can be described from any center, with any radius. 5. A figure can be moved unaltered to a new position. It is assumed in these postulates that we have at o“ disposal, (1) a plane extending indefinitely in all dire tions, (2) some means of causing a marking point to move in a straight line in any given direction, (3) some means of causing a marking point to move in that plane so as always to remain at a given distance from a given point in it. The plane may be represented by a blackboard or flat piece of paper; the second requirement is met by the use of a straightedge and marking point; the third, by a pair of compasses, or other device. By means of Post. 5, containing the principle of superposition, we are enabled to apply the criterion of Art. 14 in order to prove the equality of two given magnitudes. Thus two straight lines would be proved equal by placing them so as to coincide end with end; two angles, by causing their sides to coincide; and so on. METHOD OF PROOF. * In general, the statement and proof of a proposition consist of several distinct parts; the enunciation, the construction, and the demonstration. 1. The GENERAL ENUNCIATION or statement consists of an hypothesis (or supposition) and a conclusion. Thus in Prop. I. we have, though in different words: HYPOTHESIS. If a line is perpendicular to a second line at a certain point, *To be read in connection with Prop. I. CONCLUSION. No other line in the same plane can be perpendicular to the second line at that point. 2. The PARTICULAR ENUNCIATION, again, refers us to a particular figure or figures fulfilling the given conditions. The hypothesis and conclusion of the particular enunciation will be distinguished by the headings Given, and To Prove, respectively. 3. In the CONSTRUCTION we apply the postulates, or problems that have been proved possible, to make such changes in the form or position of the given figures as may be needful for the demonstration. 4. In the DEMONSTRATION We deduce, by a train of reasoning, the proposition to be proved, from other propositions already proved or granted. Thus, in Prop. I., we show, by means of Ax. 8 and Ax. 1 and the definition of right angles, that the angles formed by the line AE with BD must be unequal, and therefore cannot be right angles. In problems, the construction, not always a necessity in the proof of a theorem, is the essential part. Instead of the heading To Prove, however, we put the heading Required, as indicating what is required to be done. QUESTIONS. 1. What kind of a surface and what instruments are assumed as necessary in the constructions of plane geometry? 2. How do you draw a straight line longer than your straightedge? What property of straight lines do you apply? 3. With what instrument do you lay off on a line AB a shorter line CD? C B D 4. Draw a line equal to the given line AB; produce it to E, so that BE shall be equal to the line CD: AE is the sum, and BE the difference, of what lines? |