Question

If in two triangles ,sides of one triangle are proportional to the sides of other triangle ,then their corresponding angles are equal and hence the two triangles are similar
Prove . Theorem 6.4

Answer 1

by using basic proportionality theorm

Answer 2

Proved below.

Step-by-step explanation:

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.