In Fig. 6.56, PS is the bisector of ∠QPR of Δ PQR. Prove that

Construct a line segment RT parallel to SP which intersects the extended line segment QP at point T

Given: PS is the angle bisector of QPR.

Proof:

QPS = SPR (i)

By construction,

SPR = PRT (As PS || TR, By interior alternate angles) (ii)

QPS = QTR (As PS || TR, By interior alternate angles) (iii)

Using these equations, we get:

PRT = QTR

PT = PR

By construction,

PS || TR

Basic Proportionality Theorem: If a line is parallel to a side of a triangle which intersects the other sides into two distinct points, then the line divides those sides in proportion.

By using basic proportionality theorem for ΔQTR,

Hence, Proved

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