PART II. PARALLEL STRAIGHT LINES AND PARALLELOGRAMS. G BEFORE commencing this Part of Euclid the Definitions of Parallel straight lines and of a Parallelogram, and the 12th Axiom should be learnt. If a straight line EF fall on E two straight lines AB and CD, it will make eight angles, four A В angles being outside the lines AB and CD, and four angles C D within the lines AB and CD. The H angles outside AB and CD are called exterior angles ; and the angles within AB and CD are called interior angles. The angles AGH and GHD are called alternate angles. The angles BGH and GHC are also alternate angles. Euclid's 12th Axiom asserts that if the angles BGH, GHD are together less than two right angles, the lines AB and CD will meet towards B and D. By Prop. 13 the angles AGH, BGH are together equal to two right angles; and also the angles GHD, GHC are together equal to two right angles. Therefore the four interior angles AGH, BGH, GHD, GHC are together equal to four right angles. If therefore the two angles BGH, GHD are together less than two right angles, the two angles AGH, GHC will be together greater than two right angles. According to the 12th Axiom therefore the lines AB and CD will meet towards B and D when the interior angles AGH, GHC are greater than the angles BGH, GHD. But if the angles BGH, GHD are together just equal to two right angles, the angles AGH, GHC will also be together equal to two right angles ; and the angles BGH, GHD will be together equal to the angles AGH, GHC. In this case we might expect the lines AB and CD not to meet, either towards B, D or A, C; in other words, AB and CD should be parallel. They are proved parallel by Euclid in Prop. 28. PROP. 27.-- Theorem.-If a straight line, falling on two other straight lines in the same plane, make the alternate angles equal to one another, these two straight lines shall be parallel, Let the straight line EF, falling on the two straight lines AB, CD in the same plane, make the angle AGH equal to the alternate angle GHD; then AB shall be parallel to CD. A А. G B -K D H F F Proof.-If AB and CD be not parallel, they will meet, if produced, either towards B and D or towards A and C. If possible, let AB and CD, when produced, meet towards B and D in the point K. Then KGH is a triangle, having the side KG produced to A, therefore the exterior angle AGH is greater than the interior opposite angle GHD. Prop. 16 But the angle AGH is equal to the angle GHD, Hyp. therefore the angle AGH is both equal to and greater than the angle GHD, which is impossible. Therefore AB and CD cannot meet, when produced, towards B and D. Similarly it may be proved that AB and CD cannot meet, when produced, towards A and C. PROP. 28.-Theorem.-If a straight line, falling on two other straight lines, make an exterior angle equal to the interior opposite angle on the same side of the line; or, if it make two interior angles on the same side together equal to two right angles; then, the two straight lines shall be parallel. Let the straight line EF, falling on the straight lines AB, CD, make (1) the exterior angle EGB equal to the interior opposite angle GHD on the same side of the line, or (2) the two interior angles BGH, GHD on the same side together equal to two right angles ; then AB shall be parallel to CD. Proof.—(1) Because the angle EGB is equal to the angle GHD, and because the angle EGB is also equal to its vertically opposite angle AGH, but these are alternate angles ; Prop. 15 Prop. 13 (2) Because BGH, GHD are together equal to two right angles, and because the angles BGH, AGH are together equal to two right angles, therefore the angles BGH, AGH are together equal to the angles BGH, GHD. away the common angle BGH, but these are alternate angles; PROP. 29.-Theorem.--If a straight line fall on two parallel straight lines, then it shall make the alternate angles equal to one another, the exterior angle equal to the interior opposite angle on the same side of the line, and the two interior angles on the same side together equal to two right angles. Let the straight line EF fall on the two parallel straight lines AB, CD; then it shall make (1) the angle AGH equal to the alternate angle GHD; (2) the exterior angle EGB equal to the interior opposite angle GHD on the same side of the line; and (3) the two interior angles BGH, GHD on the same side together equal to two right angles. E Proof.—(1) If the angle AGH be not equal to the angle GHD, one of them must be the greater. add to each the angle BGH, then the angles AGH, BGH will be together greater than the angles BGH, GHD; but AGH, BGH are together equal to two right angles, Prop. 13 therefore the angles BGH, GHD are together less than two right angles, therefore by Axiom 12 AB and CD will meet, if produced, towards B and D; but AB and CD will never meet, since they are parallel. Therefore the angle AGH is not unequal to the angle GHD, that is, the angle AGH is equal to the angle GHD. Q. E. D. (2) Because the angle AGH is equal to the alternate angle GHD, Proved and because the angle AGH is equal to its vertically opposite angle EGB, Prop. 15 therefore the angle EGB is equal to the angle GHD. Q. E. D. (3) Because the angle EGB is equal to its interior opposite angle GHD, Prored add to each the angle BGH, therefore the angles EGB, BGH are together equal to the angles BGH, GHD, but the angles EGB, BGH are together equal to two right angles, Prop. 13 therefore the angles BGH, GHD are together equal to two right angles. Q. E. D. EXERCISES. 1. In the figure of Prop. 16, show that AB and FC are parallel. 2. Two straight lines AB and CD bisect each other at E, making AE equal to EB, and CE equal to ED. Join AC, CB, BD, DA and prove that the opposite sides of the figure ACBD are parallel. 3. In the figure of Prop. 29, make a list of all the angles which are equal to one another. 4. BAC is a given triangle. Prove that any triangle PAQ, which is formed by drawing PQ parallel to BC, will be equiangular to the triangle ABC. 5. Any straight line drawn parallel to the base of an isosceles triangle will cut off equal parts from the two equal sides of the isosceles triangle. 6. When two straight lines are parallel, state clearly all the inferences that can be drawn. 7. What is meant by the words “ on the same side” in Props. 28 and 29 ? |