10th math SAS similarity of triangle

Math Adda

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.