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10. Prove the first part of Prop. XXVI. with the trial hypothesis taken thus: “Let, if possible, AC be greater than DF.”
67. FURTHER REMARKS ON THE CASES OF EQUALITY OF
The propositions which have now been studied in five sections constitute one of the greater divisions of the First Book of the Elements. Its principal subject is that of TRIANGLES. The most important propositions are, in addition to XIII., those which treat of cases of equality “in all respects” between two triangles, namely IV., VIII., and XXVI.
The various sets of conditions under which this kind of equality subsists all require that three parts’ of the one triangle shall be equal to three similar ‘parts of the other, each to each. Taking Euclid's order of the cases, these parts must be, in each triangle
1. Two sides and the angle beween them-IV. 2. Two sides and the base (or the three sides)— VIII. 3. Two angles and the side adjacent to both-XXVI., 1st part. 4. Two angles and the side opposite one of them-XXVI., 2nd part.
If the learner were to try to make out other sets of three parts of a triangle, distinct from the above, he would find that there are two more; which it will be best to notice at once.
5. Two sides and an angle opposite one of them. These are insufficient, requiring a further condition.
6. Three angles. These may be easily shown to be insufficient also ; but still the case is important in its own place.
Euclid does not enter into the consideration of either of these cases before Book VI.; but we shall bestow some little notice on each of them in future exercises.
1. Construct a triangle, given two sides and the angle which they contain.
2. Construct a triangle, given two angles and the side adjacent to both. 3. Construct an isosceles triangle, given base and altitude. 4. Construct an angle equal to three-fourths of a right angle.
5. If the perpendiculars drawn from the vertices of a triangle upon the opposite sides are all equal, show that the triangle is equilateral.
EXAMINATION XXII. 1. What propositions form the first of the greater divisions of Book I.? Give the general subject of this division, and name its four most important propositions. 2. Give one enunciation which shall include those of V. and XVIII. 3. Give an enunciation which shall include VI. and XIX. 4. Show in what respect the order of Props. XVIII. and XIX. resembles that of V. and VI. 5. When a foot-passenger crosses a street slant-wise, of which proposition of Euclid does he show a practical knowledge ? 6. Enunciate a proposition which shall include IV. (first result) and XXIV. 7. Enunciate a proposition which shall include VIII. and XXV. 8. Find a particular case of XXIII. which is previously given. 9. Construct accordi to XXII., when the data are three straight lines respectively 3 inches, 2 in., and 1 in. long; and state the result. 10. In the enunciation of the first part of Prop. XXVI. state the hypothesis. 11. Enunciate the second part of the proposition, and state its hypothesis and thesis separately. 12. Show that the thesis in each case, as given in the text, contains three separate conclusions. What other conclusion might have been added with the same proof? 13. State the nature of each proof. Give trial hypothesis used in each part.
SECTION VI.-PROPOSITIONS XXVII.-XXXI.
Parallelism, 68. ANGLES MADE BY A STRAIGHT LINE CUTTING TWO OTHERS WHICH
DO NOT THEMSELVES INTERSECT.
Suppose XY a straight line cutting two others EF, GH, but not forming a triangle with them.
At each point of intersection are four angles. Denote the first set of four by A, B, C, D respectively; the second, by P, Q, R, S.
We shall, for the future, speak of
A, B, R, S as exterior angles ;
and P will be called the opposite interior angle to A on the same side of the line XY; similarly Q to B, &c.
Questions 1–14 refer to the figure of this article. 1. Divide the eight angles into two divisions, separated from each other by the line XY.
2. Name the interior angles which are on the same side of XY as A; also as B.
3. What interior angle is adjacent to A ?-to R? 4. State the angle opposite to R on the same side of the intersecting
also to s. 5. If D, Q are equal, show that B is equal to Q.
6. If A, P are equal, show that C and its alternate angle are also equal.
7. What proposition shows that C, D are together equal to two right angles ?
8. If C, Q are together equal to two right angles, show that D and its alternate angle are equal.
9. If C is equal to its alternate angle, show that A is equal to its opposite interior angle on the same side of the intersecting line.
10. If R is equal to its opposite interior angle on the same side, show that Q is equal to its alternate angle.
11. If A, P are equal, show that B, C, D are respectively equal to Q, R, S.
12. If A, Pare equal, show that either pair of interior angles which are on one side of XY are together equal to two right angles.
13. If angles D, P are together less than two right angles, show that FE, HG, when produced ever so far, will meet towards E, G. (Apply Axiom 12.)
14. If angles C, Q are together less than two right angles, show that EF, GH are not parallel. In what direction will they meet ?
15. In the figure to Prop. XXVIII., what angle is alternate with BGH ? Which is the opposite interior angle to CHF? Which are the two interior angles on the same side as A?
16. In the second figure to XXVII., which is the greater angle, EFC or GEF? Give the reference.
17. In the first figure of XXVII., if AEF, EFD are equal, will AB meet FD when produced to the right? If not, say why.
69. PROPOSITION XXVII.-THEOREM. If a straight line, falling on two other straight lines, makes the alternate angles equal to each other; these two straight lines shall be parallel.
Let the straight line EF, which falls upon the two straight lines AB, CD, make the alternate angles AEF, EFD equal to one another.
Then AB shall be parallel to CD.
For, if AB be not parallel to CD, AB and CD being produced will meet either towards A and C, or towards B and D.
If possible, let AB, CD produced meet towards B and Din G.
Then GEF is a triangle, and its exterior angle AEF is greater than the opposite interior angle EFG.
(I. 16) But the angle AEF is equal to the angle EFG; (Hyp.) hence AEF is both greater than and equal to EFG, which is impossible.
Therefore AB, CD produced do not meet towards B, D. In like manner it may be demonstrated,
that they do not meet when produced towards A, C. Therefore AB is parallel to CD.
(Def. 35) Wherefore, if a straight line, &c.
Q. E. D.
70. PROPOSITION XXVIII.-THEOREM. If a straight line falling upon two other straight lines, either makes the exterior angle equal to the interior and opposite upor
the same side of the line, or makes the interior angles upon the same side together equal to two right angles, the two straight lines shall be parallel to one another. Let the straight line EF, which falls upon the two straight
lines AB, CD, make the exterior angle EGB equal to the interior opposite
angle GHD upon the same side or make the two interior angles BGH, GHD on the same side
together equal to two right angles. Then, in either case, AB shall be parallel to OD.
In the first case, because the angle EGB is equal to the angle GHD, (Hyp.) and EGB is also equal to the angle AGH;
(I. 15) therefore AGH is equal to GHD;
(Ax. 1) and these are alternate angles. Therefore AB is parallel to CD.
(I. 27) In the second case, because the angles BGH, GHD are together equal to two right angles,
(Hyp.) and that the angles AGH, BGH are also together equal to two right angles;
(I. 13) therefore AGH and BGH are equal to BGH and GHD. (Ax. 1) Take away the common angle BGH; then the remaining angle AGH is equal to the remaining angle GHD;
(Ax. 3) and these are alternate angles. Therefore AB is parallel to CD.
(I. 27) Wherefore, if a straight line, &c.
Q. E. D.