COR. If each of two straight lines is perpendicular to a third straight line, the two straight lines are parallel to one another. Ex. 353. Prove the corollary. Ex. 354. Prove that the straight lines in fig. 83 would be parallel (i) if La=▲h, or (ii) if ▲ b+▲h=2 rt. <3. DEF. A plane figure bounded by three straight lines is called a triangle. DEF. A plane figure bounded by four straight lines is called a quadrilateral. DEF. The straight lines which join opposite corners of a quadrilateral are called its diagonals. DEF. A quadrilateral with its opposite sides parallel is called a parallelogram. Ex. 355. ABCD is a quadrilateral, its diagonal AC is drawn; prove that, if <BAC=LACD and 4DAC=LACB, ABCD is a parallelogram. PLAYFAIR'S AXIOM. Through a given point one straight line, and one only, can be drawn parallel to a given straight line. THEOREM 5. [CONVERSE OF THEOREM 4.] If a straight line cuts two parallel straight lines, (1) alternate angles are equal, (2) corresponding angles are equal, (3) the interior angles on the same side of the cutting line are together equal to two right angles. Data AB cuts the parallel st. lines CD, EF at G, H. To prove that (1) ▲ CGH = alt. GHF, Proof suppose GP drawn so that▲ PGH = GHF. • ▲ PGH = alt. ▲ GHF, .. PG is to EF. I. 4. .. the two straight lines PG, CG which pass through the point G are both || to EF. But this is impossible. Playfair's Axiom Ex. 356. Copy fig. 86, omitting the line PG. If LAGD=72°, find all the angles in the figure, giving your reasons; make a table. Ex. 357. Prove case (2) of Theorem 5 from first principles [i.e. without assuming case (1)]. Ex. 358. Prove case (3) of Theorem 5 from first principles [i.e. without assuming cases (1) or (2)]. Ex. 359. In fig. 87 there are two pairs of parallel lines; prove that the following pairs of angles are equal:—(i) b, l, (ii) ƒ, k, (iii) m, s, (iv) ƒ, h, (v) r, l, (vi) s, h, (vii) s, q, (viii) s, k, (ix) s, a, (x) g, l. [State your reasons carefully. e.g. WX, YZ are || and ST cuts them, Lq=Lf (corresponding angles).] W S cd X g\h mn Z pq 78 T V fig. 87. 8 Ex. 360. What do you know about the sums of (i) ▲3 f, g, (ii) L3 f, l, (iii) ▲3 m, n, in fig. 87? Give your reasons. Ex. 361. Draw a parallelogram ABCD, join AC, and produce BC to E; what pairs of angles in the figure are equal? Give your reasons. Ex. 362. A triangle ABC has ✩B=LC, and DE is drawn parallel to BC; prove that LADE=LAED. Ex. 363. If a straight line is perpendicular to one of two parallel straight lines, it is also perpendicular to the other. D B C fig. 88, Ex. 364. The opposite angles of a parallelogram are equal. [See Ex. 360.] Ex. 365. What is the sum of the angles of a parallelogram? Hence find the sum of the angles of a triangle. Ex. 366. If one angle of a parallelogram is a right angle, prove that all its angles must be right angles. NOTE ON A THEOREM AND ITS CONVERSE. The enunciation of a theorem can generally be divided into two parts (1) the data or hypothesis, (2) the conclusion. If data and conclusion are interchanged a second theorem is obtained which is called the converse of the first theorem. For example, we proved in 1. 4, that, if ▲ a = ▲d (data), then AB, CD are || (conclusion); in 1. 5, that, if AB, CD are || (data), then ad (conclusion). The data of 1. 4 is the conclusion of I. 5, and the conclusion of 1. 4 is the data of 1. 5; so that I. 5 is the converse of 1. 4 (and I. 4 is the converse of 1. 5). It must not be assumed that the converses of all true theorems are true; e.g. “if two angles are vertically opposite, they are equal" is a true theorem, but its converse "if two angles are equal, they are vertically opposite" is not a true theorem. Ex. 367. State the converses of the following: are they true? (i) If two sides of a triangle are equal, then two angles of the triangle are equal. (ii) If a triangle has one of its angles a right angle, two of its angles are acute. (iii) London Bridge is a stone bridge. THEOREM 6. Straight lines which are parallel to the same straight line are parallel to one another. Construction Draw a st. line cutting AB, CD, XY and forming with them corresponding ▲ s p, q, z respectively. Ex. 368. Prove I. 6 by means of Playfair's Axiom. [Suppose AB and CD to meet.] Ex. 369. Are the theorems true which you obtain (i) by substituting “perpendicular" for "parallel" in L. 6, (ii) by substituting "equal" for “parallel ” in 1. 6? |