10th SSS similarity for two triangles are similar

Two triangles are similar when their corresponding sides are proportional.

Let us prove this condition as a theorem, with the help of Thales Theorem.

Statement: Two triangles are similar to each other when their corresponding sides are proportional.

Proof: We are given DABC and DPQR such that

AB / PQ = BC / QR = AC / PR

We have to prove D ABC ~ DPQR

To prove this: On the sides PQ and QR of DPQR take points M and N such that

AB = PM and AC = PN. Join MN
Now,
 AB / PQ = AC / PR [Given]
i.e. PM / PQ = PN / PR [Because AB = PM and AC = PN]

Thus, MN || QR [Converse of Thales theorem]

So, <1 = <Q and <2 = <R [corresponding angles]

Therefore, DPMN ~ DPQR [AA similarity]

So, PM / PQ = MN / QR = PN / PR ----------------- (1) [Sides of similar triangles are proportional]

Since AB / PQ = PM / PQ [Because AB = PM]

And AB / PQ = BC / QR [Given]

Therefore,

PM / PQ = BC / QR ----------------- (2)

From (1) and (2)

MN / QR = BC / QR

MN =BC

Now, in D ABC and DPMN

AB =PM [We have constructed]

BC = MN [Proved above]

AC = PN [We have constructed]

Thus, DABC@D PMN [SSS congruency]

Thus, < A = <P, <B = <M and <C = vN

So DABC ~ DPQR [Because DABC @ D PMN and D PMN ~ D PQR]

We represent this condition of similarity as side - side similarity or SSS similarity

Two triangles are similar when one angle of a triangle is equal to an angle of other triangle and the sides making these angles are proportional.

Let us prove this as a theorem with the help of Thales theorem.