Q. 73.6( 24 Votes )
In figure 3.62, seg PT is the bisector of ∠QPR. A line through R intersects ray QP at point S. Prove that PS = PR
Answer :
Given: PT is angle bisector of ∠QPR
⇒ ∠QPT = ∠RPT
A line through R parallel to PT intersects ray QP at S
RS || PT
To Prove: PS = PR
Proof:
PT is angle bisector of ∠QPR
⇒ ∠QPT = ∠RPT
∠QPR = ∠QPT + ∠RPT
∠QPR = 2∠RPT (1)
RS || PT, PR is the transversal
So, ∠RPT = ∠PRS [Alternate interior angles] (2)
For ΔPRS ∠RPQ is the remote exterior angle.
∠PSR + ∠PRS = ∠QPR
Substituting (1) and (2) in the above equation
∠RPT + ∠PSR = 2∠RPT
⇒ ∠PSR = ∠RPT (3)
From (2) and (3)
∠PRS = ∠PSR
⇒ PS = PR [Sides opposite to equal angles are equal]
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