Congruence Theorem

ASA (Angle-Side-Angle) Congruence Theorem Theorem: Two triangles are congruent if any two angles and the inclined side of one triangle are equal to any two angles and the included side of the other triangle. Given: Two triangles, Triangle ABC and Triangle DEF in which ∠B = ∠E , BC=EF and ∠C = ∠F To Prove: Triangle ABC ≅ Triangle DEF Proof: There are three possibilities Case-I: When AB=DE [Figure 5-13 (a)]

In this case, we have AB=DE BC=EF (Given) and

∠B = ∠E (Given)

Therefore Triangle ABC ≅ Triangle DEF (By SAS congruence axiom) Case-II When AB
∠B = ∠F (Given)

Therefore Triangle ABC ≅ Triangle GEF

(SAS axiom)

So, ∠ACB = ∠GFE (Corresponding parts of congruent triangle are congruent) But, ∠ACB = ∠DFE (Given)

Therefore ∠GFE = ∠DFE . This is impossible unless G coincides with D. So AB must be equal to DE Hence, Therefore Triangle ABC ≅ Triangle DEF (By SAS axiom)

Case-III:

When AB>ED [Figure 5-13(c)]

In this case, let G be a point on ED produced such that AB=GE. Join GF. Now, in Triangle ABC and Triangle GEF , we have AB=GE (By supposition) BC=EF (Given)

∠B = ∠E (Given)

Therefore Triangle ABC ≅ Triangle GEF (SAS axiom) So, ∠ACB = ∠GFE (Corresponding parts of congruent triangles are congruent) But ∠ACB = ∠DFE (Given) ∠GFE = ∠DFE This is impossible unless G coincides with D.

So, AB must be equal to DE. Hence Therefore Triangle ABC ≅ Triangle DEF (By SAS axiom)

COROLLARY: SAA (Side-Angle-Angle) congruence criteria If any two angles and a non-included side of one triangle are equal to the corresponding angles and side of another triangle, then the two triangles are congruent.

Given: In Triangle ABC and Triangle DEF

∠A = ∠D, ∠B = ∠E and BC=EF To prove: Triangle ABC ≅ Triangle DEF Proof: By Angle Sum Property of triangles, in Triangle ABC we have

∠A + ∠B + ∠C = 180o

… (1)

Also in Triangle DEF

∠D + ∠E + ∠F = 180o

… (2)

From (1) and (2), we have ∠A + ∠ B + ∠ C = ∠ D + ∠ E + ∠ F

Since ∠A = ∠D , ∠B = ∠E

(Equals of equals are equal)

Therefore ∠C = ∠F

Now in Triangle ABC and Triangle DEF , we have

∠B = ∠E

(Given)

∠C = ∠F

(Proved above)

BC=EF

(Given)

Therefore Triangle ABC ≅ Triangle DEF (ASA congruence theorem)

__________________________