PROPOSITION XXIX. THEOREM. If a straight line fall upon two parallel straight lines, it makes the two interior angles upon the same side together equal to two right angles, and also the alternate angles equal to one another, and also the exterior angle equal to the interior and opposite upon the same side. Let the st. line EF fall on the parallel st. lines AB, CD. I. 48 BGH, GHD together-two rt. 4 s. I. 48 BGH, GHD cannot be together less than two rt. zs, for then AB and CD would meet if produced towards B and D, which cannot be, for they are parallel. Post. 6. Nor can s BGH, GHD be together greater than two rt. 4S, for then 48 AGH, GHC would be together less than two rt. 4s, I. 13. and AB, CD would meet if produced towards A and C Post. 6 which cannot be, for they are parallel, .. 4s BGH, GHD together=two rt. 4 s. II. 4s BGH, GHD together=two rt. <s, and 48 BGH, AGH together-two rt. 4s, I. 13. .. 48 BGH, AGH together= 48 BGH, GHD together, EXERCISES. 1. If through a point, equidistant from two parallel straight lines, two straight lines be drawn cutting the parallel straight lines; they will intercept equal portions of the parallel lines. 2. If a straight line be drawn, bisecting one of the angles of a triangle, to meet the opposite side; the straight lines drawn from the point of section, parallel to the other sides and terminated by those sides, will be equal. 3. If any straight line joining two parallel straight lines be bisected, any other straight line, drawn through the point of bisection to meet the two lines, will be bisected in that point. NOTE. One Theorem (A) is said to be the converse of another Theorem (B), when the hypothesis in (A) is the conclusion in (B), and the conclusion in (A) is the hypothesis in (B). For example, the Theorem I. a. may be stated thus: equal. Hypothesis. If two sides of a triangle be equal. Conclusion. The angles opposite those sides must also be The converse of this is the Theorem I. B. Cor.: equal. 1 Hypothesis. If two angles of a triangle be equal. Conclusion. The sides opposite those angles must also be The following are other instances: Postulate VI. is the converse of I. 17. I. 29 is the converse of I. 27 and 28. S PROPOSITION XXX. THEOREM. Straight lines which are parallel to the same straight line are parallel to one another. E P H Let the st. lines AB, CD be each || to EF. Then must AB be || to CD. Draw the st. line GH, cutting AB, CD, EF in the pts. O, P, Q. Then GH cuts the || lines AB, EF, The following Theorems are important. They admit of easy proof, and are therefore left as Exercises for the student. 1. If two straight lines be parallel to two other straight lines, each to each, the first pair make the same angles with one another as the second. 2. If two straight lines be perpendicular to two other straight lines, each to each, the first pair make the same angles with one another as the second. Book I.] PROPOSITION XXXI. 51 To draw a straight line through a given point parallel to a given straight line. E B Let A be the given pt. and BC the given st. line. It is required to draw through A a st. line || to BC. In BC take any pt. D, and join AD. Make DAE= L ADC. Produce EA to F. Then EF shall be to BC. I. 23. For AD, meeting EF and BC, makes the alternate angles equal, that is, ▲ EAD= ▲ ADC, .. EF is to BC. .. a st. line has been drawn through A || to BC. I. 27. Q. E. F. Ex. 1. From a given point draw a straight line, to make an angle with a given straight line that shall be equal to a given angle. Ex. 2. Through a given point A draw a straight line ABC, meeting two parallel straight lines in B and C, so that BC may be equal to a given straight line. PROPOSITION XXXII. THEOREM. If a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of every triangle are together equal to two right angles. M Let ABC be a A, and let one of its sides, BC, be produced to D. Then will LACD=LS ABC, BAC together. ▲ 8 ABC, BAC, ACB together=two rt. ¿ s. From C draw CE to AB. I. 31. ..▲s ECD, ACE togethers ABC, BAC together; .. LACD= Ls ABC, BAC together. And II. 28 ABC, BAO together= ▲ ACD, to each of these equals add ACB; then 48 ABC, BAC, ACB togethers ACD, ACB together, .. 48 ABC, BAC, ACB together two rt. 4 s. 1 I. 13. Q. E. D. Ex. 1. In an acute-angled triangle, any two angles are greater than the third. Ex. 2. The straight line, which bisects the external vertical angle of an isosceles triangle is parallel to the base. |