# 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 Î”ABC and Î”DEF

AB = DE

BC = EF

AC = DF

**To Prove**: Î”ABC â‰… Î”DEF

**Construction: ** Suppose BC is the longest side, draw EG such that EG = AB and âˆ GED = âˆ CBA. Join DG.

**SSS (Side-Side-Side) Congruence Theorem**

**Proof: **

In Î”ABC and Î”GED,

AB = GE ... (Const.)

âˆ ABC = âˆ GED ... (Const.)

BC = EF ... (Given)

âˆ´ DABC â‰… Î”GEF ... (SAS Cong. Axiom)

âˆ A â‰… âˆ G (c.p.c.t.) ... (i)

and AC = GF (c.p.c.t.) ... (ii)

Now, AB = GE ... (Const.)

AB = DE ... (Given)

âˆ´ DE = EG ... (iii)

Similarly, DF = GF ... (iv)

In Î”EDG, DE = EG ... (Proved above)

âˆ´ âˆ 1 = âˆ 2 ...... (âˆ s opposite to equal sides) ... (v)

In Î”DFG, FD = FG ... (Proved above)

âˆ´ âˆ 3 = âˆ 4 ...... (âˆ s opposite to equal sides)... (vi)

âˆ´ âˆ 1 + âˆ 3 = âˆ 2 + âˆ 4 ... (from (v) and (vi))

i.e., âˆ D = âˆ G ... (vii)

But, âˆ G = âˆ A ... (from (i))

âˆ A = âˆ D ... (viii)

In Î”ABC and Î”DEF

AB = DE ... (Given)

AC = DF ... (Given) âˆ A = âˆ D ... (From (viii))

âˆ´ Î”ABC â‰… Î”DEF ... (SAS Cong. Axiom)