Proof of s.s.s. similarity.triangles

Proof of S S S similarity:

Two triangles are similar if  for each angle of one triangle , there is an equal angle in another triangle. And they are the corresponding angles .

The side opposite to the corresponding angles are called corresponding sides in the  two triangles.

The  corresponding sides of two similar triangles bear the same ratio.

Proof: Let ABC and A'B'C' be two similar triangles. Since they are similar, we can assume that angles A=A', B = B' and C=C'.Let us assume again without loss of generality that ABC is the bigger triangle.

Construction: With compass take A'B' as radius  and A as centre mark B1 on AB where AB1 = A'B' . Simalilarly mark C1  on AC such that AC1 = A'C'

Now the triangles AB1C1 and A'B'C'  are congruent as

AB1=A'B' and AC1 =A'C'  by construction. Angle A=A' being equal angles of similar triangles. So,  SAS postulate holds good for congruency.

Therefore, the angles AB1C1 =  B' = B  and angle AC1B1 =C' = C

This confirms that the line  B1C1  is  parallel  to  the line or side BC.

The line B1C1 is parallel to BC, B1 being on AB and C1 beeing AC , the ratio, AB1/ AB = AC1/AC  = B1C1/BC  by Thales intercept theorem.

The converse is also true. That is, if the corresponding sides of two triangles are in the same ratio, then  the triangles are similar or the corresponding angle are equal. The proof involves, identifying the smaller triangle A'B'C' and AB1C1 and applying the Thales intercept theorem and prove that B1C1 is parallel as AB1/AC= AC1/AC. Thus the Parallel property of B1C1 with BC gives us the corresponding angles AB1C1 = B =B'and AC1B1 = C=C'. Consequently the remaining angle A has to be equal to A'.

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