Theorem: If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles included by those sides equal, then the triangles are equal in all respects.
Construction: Let ABC, DEF be two triangles in which 
and
and the included angle 
is equal to included angle 
.
It is required to prove that the
in all respects.
Proof: Apply the 
to the 
so that the point A falls on the point D; and the side AB along the side DE. Then, because so the point B must coincide with the point E. And, because AB falls along DE, and the
, so AC must fall along DF. And, because
, the point C must coincide with the point F. Then, since B coincides with E, and C with F, hence, the side BC must coincide with the side EF. Hence, the coincides with the , and is therefore equal to it in all respects. QED.
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Nalin Pithwa
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Theorem: If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles included by those sides equal, then the triangles are equal in all respects.
Mathematics Hothouse 