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Question
Two lines l and m intersect at the point O and P is a point on a line n passing through the point O such that P is equidistant from l and m. Prove that n is the bisector of the angle formed by l and m.
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Solution
Given: Two lines l and m intersect at the point O and P is a point on a line n passing through O such that P is equidistant from l and m i.e., PQ = PR.
To prove: n is the bisector of the angle formed by l and m i.e., n is the bisector of ∠QOR.
Proof: In ΔOQP and ΔORP,
∠PQO = ∠PRO = 90° ...[Since, P in equidistant from l and m, so PQ and PR should be perpendicular to lines l and m respectively]
OP = OP ...[Common side]
PQ = PR ...[Given]
∴ ΔOQP ≅ ΔORP ...[By RHS congruence rule]
⇒ ∠POQ = ∠POR ...[By CPCT]
Hence, n is the bisector of ∠QOR.
Hence proved.
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