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In order to prove, 'The bisector of an a...

In order to prove, 'The bisector of an angle of a triangle divides the side opposite to the angle in the ratio of the remaining sides.
`(i)` Draw a neat labelled figure.
`(ii)` Write 'Given' and 'To prove'.

Text Solution

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Proof `:` In `Delta PMR `,
seg `QS || ` side `MR ` ….(Constructions )
`:.` by basic proportionality theorem,
`(PQ)/( QM) = ( PS)/( SR )` ......(1)
Ray `QS ||` side MR and line PM is the transversal.
`:. /_ PQS ~= /_ QMR ` ...[Corresponding angles ] ...(2)
Ray `QS ||` side MR and line QR is the transversal
`:. /_SQR ~= /_ QRM ` ...[Alternate angles ] ...(3)
`/_ PQS ~= /_ SQR` ..[Ray QS is the bisector of `/_ PQR ` ] ....(4)
In `Delta QRM`
`/_ QMR ~= /_QRM ` ...[From (2) , (3) and (4) ]
`:.` seg `QR ~= ` seg QM ....[Converse of isosceles triangle theorem ]
`:. (PQ )/(QR) = (PS)/(SR)` .....[From (1) and (5) ]
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