**Statement:**Two triangles are similar to each other when one angle of a triangle equal to an angle of other triangle and sides making these angles are proportional.

**Proof:**We are given DABC and DPQR such that < A = <P

And AB / PQ = AC / PR

We have to prove. DABC ~ 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

PM / PQ = PN / PR [Because AB = PM and AC = PN]

Thus, MR || QR [

**Converse of Thales theorem**]

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

Therefore, DPMN ~ DPQR [A. A Similarity]

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

Now, in DABC and DPMN

AB = PM [we have constructed]

<A = <P [Given]

AC = PN [we have constructed]

Thus, DABC@ DPMN

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

So DABC ~ DPQR [Because DABC @ DPMN and DPMN ~ DPQR]

This condition of similarity is known as SAS similarity.

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