Question

in angle pqr, ps is the bisector of angle p,s is a point such angle psr > angle psq. prove that pr> pq​

Answer 1

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Given: A triangle PQR in which PS is the internal bisector of angle P.

R.T.P: QS/SR=PQ/PR

Construction: Draw RE parallel PS to meet QP produced in E.

Proof:

Here we have,

ER || PS

Then,

angle 2 = angle 3.........1 (alternative angles)

and,

angle 1 = angle 4....... 2 (corresponding angles)

Given that PS in an angular bisector

then,we have

angle 1 = angle 2........3

By eq1 , eq2 , eq3 we get,

angle 3 = angle 4

Then,

PR=PE...........4 (sides opposite to equal angles)

In triangle QRE we have,

RE || PS

By basic proportionality theorem we get,

QS/SR = QP/PE

QS/SR = QP/PR

QS/SR = PQ/PR

Hence proved.