Question
Sun June 01, 2008 By:

prove ASA cogruency?

Expert Reply
Sun June 01, 2008
We are given two triangles ABC and DEF in which:
angle B = E, C = F 
and BC = EF
We need to prove that ABC is congruent toDEF
For proving the congruence of the two triangles we consider three cases
Case (i) :Let AB = DE
And we have,B = E (Given)
BC = EF (Given)
So, triangle ABC is congruent to DEF (By SAS rule)
Case (ii) :Let if possible AB > DE. So, we can take a point P on AB such that
PB = DE. Now consider triangles PBC and  DEF
We have,
PB = DE (By construction)
B = E (Given)
BC = EF (Given)
So, we get
PBC is congruent toDEF, by the SAS
Since the triangles are congruent, their corresponding parts will be equal.
So, angle∠ PCB = ∠ DFE
But, we are given that
Angle ACB = ∠ DFE
So, angle ACB = PCB
But, this is possible only if P coincides with A.
or, BA = ED
So, ABC is congruent to DEF (by SAS axiom)
Case (iii) :If AB < DE, we can choose a point M on DE such that ME = AB and
By similar arguments as given in Case (ii), we can conclude that AB = DE and so,
Triangle ABC is congruent toDEF.
 
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