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Question
In the given figure, seg PT is the bisector of ∠QPR. A line through R intersects ray QP at point S. Prove that PS = PR.
Solution
Given: Seg PT is the bisector of ∠QPR.
To prove: PS = PR
Construction: Draw seg SR || seg PT.
Proof:
seg PT is the bisector of ∠QPR. ...[Given]
∴ ∠QPT = ∠RPT ...(i)
seg PT || seg SR ...[Construction]
and seg QS is their transversal.
∴ ∠QPT = ∠PSR …(ii) ...[Corresponding angles]
seg PT || seg SR ...[Construction]
and seg PR is their transversal.
∴ ∠RPT = ∠PRS ...(iii) ...[Alternate angles]
∴ ∠PRS = ∠PSR …(iv) ...[From (i), (ii) and (iii)]
In ∆PSR,
∠PRS = ∠PSR ...[From (iv)]
∴ PS = PR ...[Converse of isosceles triangle theorem]
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