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If /B and /Q are acute angles such that ...

If `/_B` and `/_Q` are acute angles such that `sinB=sinQ` . Then prove that `/_B=/_Q` .

Text Solution

Verified by Experts

Let `Delta ACD and DeltaBEF` be two right triangles
such that sin `A= sin B.`
then , sin `A= sin B`
`implies (CD)/(AD)=(EF)/(BF)`
`implies (CD)/(EF)=(AD)/(BF)and AD=k(BF).`
`therefore (AC)/(BE)=(sqrt(AD^(2)-CD^(2)))/(sqrt(BF^(2)-EF^(2)))" " [ " using pythagoras' theorem "]`
`=(ksqrt(BF^(2)-EF^(2)))/(sqrt(BF^(2)-EF^(2)))=K["using (i) "].`
`therefore (CD)/(EF)=(AD)/(BF)=(AC)/(BE).`
`implies Delta ACD ~ Delta BEF ` and hence `angle A=angle B.`
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