(e).. by Euc. I. 5, angle BDC is equal to angle B CD. But we have proved that angle BDC is greater than angle B CD (d). .. If A C and AD are equal, and at the same time B C and BD are equal, then angle B DC is both greater than (d) and equal to (e) angle B C D. But this is impossible: and so it is impossible for AC and AD to be equal, at the same time that BC and BD are equal. General Enunciation (CASE III). Q. E. D. Given. The two triangles CAB, DAB, as before, but having the vertex of one triangle on a side of the other. Required. To prove the same as in Cases I., II. Proof. The truth of the proposition in this case is selfevident. For DA is a part of CA, and the whole is greater than its part. EXERCISE.-On any base, construct two triangles, one on each side of the base, having the two sides terminated in one extremity equal, and likewise those terminated in the other extremity equal. (Use circles, and prove as in Euc. I. 22.) 100 PART III. THE THEORY OF PARALLELS. Definition of Parallel Straight Lines. Parallel straight lines are such as being produced ever so far, both ways, do not meet. Thus, A B and CD are parallel lines, they are every where the same dis A B D tance apart; and if produced to any length, still remain every where the same distance apart. Euclid's twelfth Axiom. If a straight line meet two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles, those straight lines being continually produced, shall at length meet on that side on which are the angles which are less than two right angles. The line EH is said to fall on the two lines A B and CD. But the line FG is said to meet the same two lines. As will be seen; the line FG, meeting the lines AB, CD makes four angles;-A FG, FGC on one side of FG, and B FG, FGD on the other side of FG. These angles are called, interior angles. Euclid assumes as a self-evident truth, that if either pair of angles be less than two right angles, then the lines must meet if produced, and meet on that side on which the angles are less than two right angles. That is to say, if the angles BFG, FGD are together less than two right angles, then AB and CD being produced in the direction of B and D will meet. Let us produce the lines A B and CD till they meet at K (Fig. 2). Then we shall see the truth of the assumption, and the reason why they must meet if angles BFG and FGD are less than two right angles. For KFG is a triangle. And, by Euc. I. 17, any two angles of a triangle are less than two right angles. So that the lines EB and G D are only parts of the two sides of a triangle produced. Hence we see that when a straight line meets two other straight lines, and makes the two interior angles on the same side less than two right angles, those lines will meet if produced because they are really parts of the sides of a triangle. ALTERNATE AND EXTERIOR ANGLES. As we have just learnt, angles AFE, FEC, BFE, FED, are called interior angles. Of those A FE A B E D H and FED are called alternate angles, and so are BFE and EFC. The four angles AFG, GFB, CEH, HED, are called exterior angles. The interior angle FED is said to be opposite to the exterior angle GFB. So also FE C is opposite to GFA, THEOREM (Euclid I. 27). Repeat. The enunciation of Euc. I. 16, and the definition of parallel straight lines. General Enunciation. If a straight line, falling on two other straight lines, make the alternate angles equal to each other, the two straight lines shall be parallel to each other. |