Ex. 333. From a point A in a straight line AB, straight lines AC and AD are drawn at right angles to AB on opposite sides of it; prove that CAD is a straight line. Ex. 334. From a point O in a straight line AOC, OB and OD are drawn on opposite sides of AC so that LAOB=LCOD; prove that BOD is a straight line. B fig. 80. Ex. 335. Three straight lines OB, OA, OC are drawn from a point (see fig. 78), OP bisects BOA, OQ bisects' LAOC; prove that, if POQ is a right angle, BOC is a straight line. Ex. 336. Two straight lines XOX', YOY' intersect at right angles; OP bisects LXOY, OQ bisects X'OY'. Is POQ a straight line? [Find the sum of 48 POY, YOX', X'OQ.] Revise Ex. 64-66. Ex. 337. If a straight line rotates about its middle point, do the two parts of the straight line turn through equal angles? If the line rotates about any other point, are the angles equal? Ex. 338. ABCD are four points in order on a straight line; if ACBD then ABCD. Ex. 339. If two straight lines AOB, COD intersect at O (see fig. 81) what is the sum of 4 AOB, BOC? What is the sum of LS BOC, COD? DEF. The opposite angles made by two intersecting straight lines are called vertically opposite angles (vertically opposite because they have the same vertex). B fig. 81. Ex. 340. Name two pairs of vertically opposite angles in fig. 81. THEOREM 3. If two straight lines intersect, the vertically opposite angles are equal. Data Two st. lines intersect forming the angles x, y, z, w. To prove that Proof Revise Ex. 67, 68. Lxvert. opp. 42, Lyvert. opp. w. 4x+2y=2rt. 4s, (Why?) .. L x = LZ. Sim1y Ly=LW. Ex. 341. Write out in full the proof that Ly=Lw in 1. 3. Q. E. D. Ex. 342. Draw a triangle and produce every side both ways; number all the angles in the figure, using the same numbers for angles that are equal. Ex. 344. If two straight lines AOC, BOD intersect at O and OX bisects LAOB, then XO produced bisects COD. Ex. 345. The bisectors of a pair of vertically opposite angles are in one and the same straight line. PARALLEL STRAIGHT LINES. DEF. Parallel straight lines are straight lines in the same plane, which do not meet however far they are produced in either direction. DEF. In the figure two straight lines are cut by a third straight line; 4 c and ƒ are called alternate angles, 8 b and ƒf corresponding angles (sometimes L sband ƒ are spoken of as L 66 an exterior angle and the interior opposite angle on the same side of the cutting line"). a/b c/d f e g/h fig. 83. Ex. 346. Name another pair of alternate angles in fig. 83. Ex. 347. Ex. 348. (i) c, f, (ii) b, f, What are the names of the following pairs: (iii) h, d, (iv) a, d, (v) c, g, (vi) e, f, (vii) e, a, (viii) e, d? Ex. 349. Prove that if a straight line cuts two other straight lines and makes a pair of alternate angles equal, then a pair of corresponding angles are equal. [That is, in fig. 83, prove that if LcLf, then ▲ b=Lƒ.] Ex. 350. In fig. 83, prove that, if LcLf, then Ld+Lf=2rt. 45. State this formally as in Ex. 349. (4 d and ƒ are interior angles on the same side of the cutting line.) Revise Ex. 167. Ex. 351. Draw two parallel straight lines and a line cutting them; measure a pair of alternate angles. Ex. 352. Take a strip of paper about two inches wide with parallel sides, cut it across as in fig. 84; measure the angles so formed with your protractor, noting which are equal, and test whether the two pieces can be made to coincide (i.e. fit on one another exactly). THEOREM 4.* (1) When a straight line cuts two other straight lines, if a pair of alternate angles are equal, then the two straight lines are parallel. d B A' F fig. 84. (1) Data The st. line EF cuts the two st. lines AB, CD at E, F, forming the s a, b, C, Take up the part AEFC, call it A'E'F'C'; and, turning it round in its own plane, apply it to the part DFEB so that E' falls on F and E'A' along FD. * The proof of this theorem should be omitted at a first reading. Now if EB and FD meet when produced towards B and D, F'C' and E'A' must also meet when produced towards C' and A', i.e. FC and EA must also meet when produced towards C and A. .. if AB, CD meet when produced in one direction, they will also meet when produced in the other direction; but this is impossible, for two st. lines cannot enclose a space. .. AB, CD cannot meet however far they are produced in either direction. .. AB and CD are parallel. Q. E. D. When a straight line cuts two other straight lines, if (2) a pair of corresponding angles are equal, or (3) a pair of interior angles on the same side of the cutting line are together equal to two (2) Data The st. line GH cuts the two st. lines AB, CD forming the s a, b, c, d, e. Le= corresp. 4 d. To prove that AB, CD are parallel. |