If in two trangles,sides of one …

Given:

Here  in two triangles, sides of one triangle are proportional to the sides of other triangle.

Also their corresponding angles are equal and hence the two triangles are similar.

Construction:

Let us draw lines at E and F making the angles as marked in the diagram and meeting at G.

Proof:

Now, ΔABC and ΔGEF are equiangular and hence similar and so corresponding sides are in proportion.

         AB : BC = GE : EF

But    AB : BC = DE : EF          (Given)

So              GE = DE                  (1)

 Similarly,  AC : CB = GF : FE

But    AB : BC = DF : FE    (Given)

So              GF = DF                  (2)

 EF is common to both triangles DEF and GEF.          (3)

So from Eq (1), (2) and (3) we have

ΔDEF ≅ ΔGEF.

Therefore triangles ABC and DEF are equiangular and hence similar etc.

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