Congruence Theorem

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: Triangle ABC and Triangle DEF such that AB=DE, BC=EF and AC=DF To Prove: Triangle ABC ≅ Triangle DEF Construction: Suppose BC is the longest side. Draw EG such that ∠FEG = ∠ABC and EG=AB. Join GF and GD. Proof: In Triangle ABC and Triangle GEF , we have BC=EF (Given) AB=GE (By construction) and ∠ABC = ∠FEG (By construction

Therefore By SAS criterion of congruence

Triangle ABC ≅ Triangle GEF

Hence, ∠A = ∠G and AC=GF (Corresponding parts of congruent triangles are congruent) Now, AB=DE and AB=GE So DE=GE

… (1)

Similarly, AC=DF and AC=GF So DF=GF

(Equals of equals are equal)

… (2)

Now, in Triangle EGD , we have DE=GE [From (1)] So, ∠EDG = ∠EGD

… (3)

In Triangle FGD , we have DF=GF [From (2)] So, ∠FDG = ∠FGD

… (4)

From (3) and (4), we have

∠EDG + ∠FDG = ∠EGD + ∠FGD Therefore ∠D = ∠G But ∠G = ∠A

(Proved above)

Therefore ∠A = ∠D

(Transitive Property)

… (5)

Now, in Triangle ABC and Triangle DEF , we have AB=DE

(Given)

∠A = ∠D

[from (5)]

and AC=DF (Given) So, by SAS criterion of congruence Triangle ABC ≅ Triangle DEF

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