SSS
We proved that SAS implies the Pons Asinorum, or the isosceles triangle theorem. This worksheet shows that SAS further implies SSS.
SAS implies SSS
Proof:
Let ΔABC and ΔABC' be two triangles satisfying the SSS criterion after one has been flipped and placed so that two sides coincide on segment AC. It follows that AB=AB' and CB=CB'.
By using the Pons Asinorum twice, we get that the green angles are congruent to each other and also the red. Angle addition yields that ABC is anticongruent to AB'C.
SAS then implies ΔABC and ΔAB'C are anticongruent and hence that the original triangles were congruent before flipping.