. Elements of geometry : containing books I to III. ; .. l A DC is greater than z BCD ;much more is L BDC greater than l , v BC=BD, .: l BDC= z BCD,that is, z BDC is both equal to and greater than z BCD;.vhich is absurd. Secondly, when the vertex D of one of the as falls withinthe other A (Fig. Produce AC and AD to E and FThen v AC= AD. .-. z ECD = z i^DC. I. 5. But z £CD is g! eater than z JB( !D ; .-. z FD& i> greater than z JSCZ> ;much more is z _BZ>C greater than z . : BC=BD, .. l BD(= 1 BCD ;that is, l BDC is both equal to and greater than cBCD\which is absurd.
. Elements of geometry : containing books I to III. ; .. l A DC is greater than z BCD ;much more is L BDC greater than l , v BC=BD, .: l BDC= z BCD,that is, z BDC is both equal to and greater than z BCD;.vhich is absurd. Secondly, when the vertex D of one of the as falls withinthe other A (Fig. Produce AC and AD to E and FThen v AC= AD. .-. z ECD = z i^DC. I. 5. But z £CD is g! eater than z JB( !D ; .-. z FD& i> greater than z JSCZ> ;much more is z _BZ>C greater than z . : BC=BD, .. l BD(= 1 BCD ;that is, l BDC is both equal to and greater than cBCD\which is absurd. Lastly, when the vertex D of one of the l s falls on aMde ]>< of the other, it is plain that BC and iiX» cannotbe equal q. e. D. H2 EUCLIDS ELEMENTS. [Books I. & IL Note 14. Euclid!$ Proof of I, 8. If tirn triangles have two tides of the one equal to twotides of the other, each to each, and hare likewise Iequal, the angle which is contained by the two titles of themust be equal to the angle contained by the two sides oftfu Let the sides of the as ABC, DEF be equal, each to each,that is, AB=DE, AC=DF and BC=EF. Then mud l BAC= i EDF. Apply the a A B( to the a DEF. so that pt. B is on pt E, and BC on V BC=EF, .*. C « ill coincide with /.and BC will coincide with EF. Then 42? and 4(7 must coincide with DE and DF. For if AB and AO have a different position, as QE, CF,then upon the ie and the Bame aide of it there can be two L b, which have their Bides which are terminated inone extremity f the base equal, and their aides which an initiated ii the Other extremity of the base also equal : whichLb impossible. I. 7. .•. since l>a~.- BC coincides with base EF, .1 /. must coincide with DE, and A( with T>F;.•. l BACco\ml\> s with and is equal to i EDF. q. k. D. Book I. & II.] AXOTHER PROOF OF T. 24. 3 Note 15. Another Proof of I. 24. In the as ABC, DEF, let AH=DE and AC=DF,and let z !).-!( lie greater than _ A7i\17 •/( //(Ko>C be tj> Jc
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