5. A Corollary is a statement the truth of which follows readily from an established proposition; it is therefore appended to the proposition as an inference or deduction, which usually requires no further proof. 6. The following symbols and abbreviations are used in the text of this book. par After Part I. pt. for point, perp. for perpendicular, st. line straight line, parallelogram, rt. L right angle, rectil. rectilineal, par (or II) parallel, circle, sq. square, circumference; and all obvious contractions of commonly occurring words, such as opp., adj., diag., etc., for opposite, adjacent, diagonal, etc. ce (For convenience of oral work, and to prevent the rather common abuse of contractions by beginners, the above code of signs has been introduced gradually, and at first somewhat sparingly.] In numerical examples the following abbreviations will be used. m. for metre, cm. for centimetre millimetre. km. kilometre. Also inches are denoted by the symbol ("). Thus 5" means 5 inches. mm. ON LINES AND ANGLES. THEOREM 1. [Euclid I. 13.] The adjacent angles which one straight line makes with another straight line on one side of it, are together equal to two right angles. Let the straight line co make with the straight line AB the adjacent < AOC, COB. It is required to prove that the Ľ AOC, COB are together equal to two right angles. Suppose OD is at right angles to BA. Proof. Then the ĽAOC, COB together = the three _ AOC, COD, DOB. Also the 2" AOD, DOB together = the three 4AOC, COD, DOB. :, the AOC, COB together= the Ľ AOD, DOB =two right angles. Q.E.D. = PROOF BY ROTATION. Suppose a straight line revolving about O turns from the position OA into the position oC, and thence into the position OB; that is, let the revolving line turn in succession through the 28 AOC, COB. Now in passing from its first position OA to its final position OB, the revolving line turns through two right angles, for AOB is a straight line. Hence the L AOC, COB together=two right angles D COROLLARY 1. If two straight lines cut one another, the four angles so formed are together equal to four right angles. А. B For example, LBOD+DOA+ LAOC + LCOB=4 right angles. B COROLLARY 2. When any number of straight lines meet at a point, the sum of the consecutive angles so formed is equal to four right angles. A 'E For a straight line revolving about O, and turning in succession through the Z8 AOB, BOC, COD, DOE, EOA, will have made one complete revolution, and therefore turned through four right angles. DEFINITIONS. (i) Two angles whose sum is two right angles, are said to be supplementary; and each is called the supplement of the other. Thus in the Fig. of Theor. I the angles AOC, COB are supplementary. Again the angle 123° is the supplement of the angle 57°. (ii). Two angles whose sum is one right angle are said to be complementary; and each is called the complement of tho other. Thus in the Fig. of Theor. 1 the angle DOC is the complement of the angle AOC. Again angles of 34° and 56° are complementary. COROLLARY 3. (i) Supplements of the same angle are equal. (ü) Complements of the same angle are equal THEOREM 2. [Euclid I. 14.] If, at a point in a straight line, two other straight lines, on opposite sides of it, make the adjacent angles together equal to two right angles, then these two straight lines are in one and the same straight line. At 0 in the straight line co let the two straight lines OA, OB, on opposite sides of co, make the adjacent 2 AOC, COB together equal to two right angles : (that is, let the adjacent LAOC, COB be supplementary). It is required to prove that OB and OA are in the same straight line. Produce AO beyond o to any point X: it will be shewn that OX and OB are the same line. Proof. Since by construction AOX is a straight line, .. the - COX is the supplement of the L COA. Theor. 1. But, by hypothesis, the - COB is the supplement of the L COA. ... the COX = the L COB; .: OX and OB are the same line. But, by construction, ox is in the same straight line with OA; hence OB is also in the same straight line with OA. Q.E.D. EXERCISES. :1, Write down the supplements of one-half of a right angle, four. thirds of a right angle; also of 46°, 149°, 83°, 101° 15'. 2. Write down the complement of two-fifths of a right angle ; also of 27°, 38° 16', and 41° 29' 30". 3. If two straight lines intersect forming four angles of which one is known to be a right angle, prove that the other three are also right angles. 4. In the triangle ABC the angles ABC, ACB are given equal. If the side BC is produced both ways, shew that the exterior angles so formed are equal. 5. In the triangle ABC the angles ABC, ACB are given equal. If AB and AC are produced beyond the base, shew that the exterior angles so formed are equal. DEFINITION The lines which bisect an angle and the adjacent angle made by producing one of its arms are called the internal and external bisectors of the given angle. Thus in the diagram, OX and OY are the internal and external bisectors of the angle AOB. с A 6. Prove that the bisectors of the adjacent angles which one straight line makes with another contain a right angle. That is to say, the internal and external bisectors of an angle are at right angles to one another. 7. Shew that the angles AOX and COY in the above diagram are complementary. 8. Shew that the angles BOX and COX are supplementary; and also that the angles AOY and BOY are supplementary. 9. If the angle AOB is 35°, find the angle COY. |