Page:First six books of the elements of Euclid 1847 Byrne.djvu/38

4 F two triangles have two sides of the one respectively equal to two sides of the other, ( to and  to ) and the angles ( and ) contained by those equal sides also equal; then their bases or their sides ( and ) are also equal: and the remaining and their remaining angles opposite to equal sides are respectively equal ( =  and  = ): and the triangles are equal in every respect.

Let the two triangles be conceived, to be so placed, that the vertex of the one of the equal angles, or ; shall fall upon that of the other, and  to coincide with, then will   coincide with  if applied: consequently  will coincide with , or two straight lines will enclose a space, which is impossible (ax. 10), therefore  = ,  =  and =, and as the triangles  and  coincide, when applied, they are equal in every respect. Q.E.D.