The Public School Euclid and Algebra . 14 Euclids elements. PROPOSITION 4. Theorem. If two triangles have ttvo sides of the one eqiud to two sides ofthe othei\ each to each^ and have also the angles containedhy those sides eqiiaX to each other^ they slwXl also have theirthird sidss equal; and the t^vo triangles shall he equals andthe other angles shall he equal, each to each, Jiamely thoseto tvhich the equal sides are oj^ In the AS ABC and DEF,let AB = DE,AG = DF,and Z BAG = / EDF,It is required to prove that BG = EF, A ABG = A DEF,I ABG = Z DEF,and Z AGB = Z DEE,If A ABG be apphed to
The Public School Euclid and Algebra . 14 Euclids elements. PROPOSITION 4. Theorem. If two triangles have ttvo sides of the one eqiud to two sides ofthe othei\ each to each^ and have also the angles containedhy those sides eqiiaX to each other^ they slwXl also have theirthird sidss equal; and the t^vo triangles shall he equals andthe other angles shall he equal, each to each, Jiamely thoseto tvhich the equal sides are oj^ In the AS ABC and DEF,let AB = DE,AG = DF,and Z BAG = / EDF,It is required to prove that BG = EF, A ABG = A DEF,I ABG = Z DEF,and Z AGB = Z DEE,If A ABG be apphed to A DEF, so that A falls on D, andAB falls on DE, then B will coincide with E, because AB = DE. ^^yp- And because AB coincides with DE, and Z BAG=l EDF, Hyp. .-. ^C will fall on Di^.And because AG = DF, ^^yP .. G will coincide with because B coincides with E, and G with F^.. BC will coincide with , if not, let it fall otherwise, as EGF ETTCLrnS ELEMENTS. 15 Then tlie tw(j straight lines i/C and EF will enclose a space,whicli is impossible. Ax. 10. Hence JiC coincides with and .*. = El\ Ax. 8 \ ABC .-. = 1 J)KJ\ Z ABC - . / DEF, and Z ACB .-. = z />/i^- Questions on Proposition 4. 1. State the axioms used in the proof. 2. Are the words, each to each, necessary in the enunciation ? 3. Does ^ 6 necessarily fall on DF^ if AB coincides with DEi 4. Prove the proposition, beginning the superposition by applying B
Size: 2865px × 872px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1800, bookdecade1890, bookidpublicschool, bookyear1894
