Through a given point P to construct a straight line parallel to a given straight line AB. Construction Through P draw a straight line PQ meeting AB in Q. With centre Q and any convenient Then PG is to BA. Proof Join DE, GF. In the As GPF, DQE, GP = DQ, FP = EQ, FG = ED, G P F E Q B Fig. 31. .. the As are congruent and ▲ P = alt. ▲ DQE, .. PG is || to QD. Constr. Th. 4 (i). Ex. Through P draw a parallel to a line AB by a construction depending on the equality of corresponding angles. For Exercises on Constructions see Practical Geometry, Ch. 3. CHAPTER V INEQUALITIES THEOREM 16. If two sides of a triangle are unequal, the greater side has the greater angle opposite to it. B Fig. 32. Data To prove that ABC is a triangle in which AB > AC. LACBL B. Construction From AB, the greater side, cut off AD = AC. Proof Join CD. In AACD, AD = AC, .. LACD = 4 ADC. Th. 12. Th. 8, Cor. 1. But since the side BD of the ▲ DBC is produced to A, .. ext. ▲ ADC > int. opp. ▲ B, .. 4 ACD > LB. But ACB > its part ▲ ACD, THEOREM 17 [CONVERSE OF THEOREM 16]. If two angles of a triangle are unequal, the greater angle has the greater side opposite to it. 8 Fig. 33. A Data To prove that ABC is a triangle in which ▲ C> < B. AB > AC. Either (i) AB > AC, or (ii) AB = AC, or (iii) AB < AC. If, as in (iii), AB < AC, then CLB (Th. 16), which is impossible. Data. NOTE. The method of proof used in Th. 17 is called the method of exhaustion. The different possibilities are exhausted, and that stated in the theorem is shown to be the only one tenable. Observe that Th. 17 is deduced from Th. 16, and therefore that Th. 16 must not be proved by assuming Th. 17. It is necessary to remember which theorem comes first. THEOREM 18. Any two sides of a triangle are together greater than the third side. "It was the habit of the Epicureans, says Proclus, to ridicule this theorem as being evident even to an ass and requiring no proof, and their allegation that the theorem was 'known' even to an ass was based on the fact that, if fodder is placed at one angular point and the ass at the other, he does not, in order to get to his food, traverse the two sides of the triangle, but only the one side separating them (an argument which made Savile exclaim that its authors were 'digni ipsi, qui cum Asino foenum essent'). Proclus replies truly that a mere perception of the truth of the theorem is a different thing from a scientific proof of it and a knowledge of the reason why it is true. Moreover, as Simson says, the number of axioms should not be increased without necessity" (T. L. Heath, Euclid's Elements, vol. i, p. 287). THEOREM 19. Of all the straight lines that can be drawn to a given straight line from a given point outside it, the perpendicular is the shortest. Data AB is a straight line and O a point outside it; ON is drawn 1 to AB meeting it in N. Sim' ON may be proved less than any other st. line drawn from O to meet AB. .. ON is the shortest of all such lines. NOTE. The shortest distance of a point from a straight line is called for brevity the distance of the point from the line. Ex. 1. In a right-angled triangle, the hypotenuse is the longest side. Ex. 2. The side opposite the obtuse angle of an obtuse-angled triangle is the greatest side. Ex. 3. If one angle of a triangle is known to be the greatest angle, the side opposite to it must be the greatest side. Ex. 4. The angles at the ends of the greatest side of a triangle are acute. Ex. 5. In a parallelogram ABCD, AB>AD; prove that <ADB><BDC. [What angle is equal to ▲ BDC?] Ex. 6. In a quadrilateral ABCD, AB is the shortest side and CD is the longest side; prove that <B>LD, and ▲A>LC. [See fig. 36. Draw a diagonal.] Ex. 7. Assuming that the diagonals of a parallelogram ABCD bisect one another, prove that, if BD>AC, then LDAB is obtuse. Ex. 8. Prove Th. 16 by means of the following construction:--from AB cut off AD=AC, bisect LBAC by AE, join DE. (See fig. 37.) Ex. 9. AE bisects the angle BAC of a triangle and meets the base BC at E. Prove that AB is greater than BE and AC greater than EC. B E A Fig. 37. |