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SSS (Side-Side-Side) Congruence Theorem

Theorem: Two triangles are congruent if the three sides of one triangle are equal to the corresponding three sides of the other triangle.

Given: In ΔABC and ΔDEF
AB = DE
BC = EF
AC = DF

To Prove: ΔABC ΔDEF
Construction: Suppose BC is the longest side, draw EG such that EG = AB and GED = CBA. Join DG.
 

SSS (Side-Side-Side) Congruence Theorem




Proof:
In ΔABC and ΔGED,
       AB = GE ... (Const.)

    ABC = GED ... (Const.)
        BC = EF ... (Given)

∴ DABC ΔGEF ... (SAS Cong. Axiom)
A G (c.p.c.t.) ... (i)
and AC = GF (c.p.c.t.) ... (ii)
Now, AB = GE ... (Const.)
AB = DE ... (Given)

DE = EG ... (iii)
Similarly, DF = GF ... (iv)
In
ΔEDG, DE = EG ... (Proved above)
1 = 2 ...... (s opposite to equal sides) ... (v)
In
ΔDFG, FD = FG ... (Proved above)
3 = 4 ...... (s opposite to equal sides)... (vi)
1 + 3 = 2 + 4 ... (from (v) and (vi))
i.e., D = G ... (vii)
But, G = A ... (from (i))
A = D ... (viii)
In ΔABC and ΔDEF
AB = DE ... (Given)
AC = DF ... (Given) A = D ... (From (viii))
ΔABC
≅ ΔDEF ... (SAS Cong. Axiom)

 





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