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Standard IX

Mathematics

RHS Criteria for Congruency

Question 15Tw...

Question

Question 15
Two lines I 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 I and m. Prove that n is the bisector of the angle formed by I and m.

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Solution

In $Δ O Q P a n d Δ O R P,$
$∠ P Q O = ∠ P R O = 90^{\circ}$
[since, P in equidistant from I and m, so PQ and PR should be perpendicular to lines I and m respectively]
OP = OP [comon side]
$∴ Δ O Q P ≅ Δ O R P$ [by RHS congruence rule]
$\Rightarrow ∠ P O Q = ∠ P O R$ [by CPCT]
Hence, n is the bisector of $∠ Q O R .$

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RHS Criteria for Congruency

Standard IX Mathematics

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Question 15 Two lines I 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 I and m. Prove that n is the bisector of the angle formed by I and m.

In ΔOQP and ΔORP, ∠PQO=∠PRO=90∘ [since, P in equidistant from I and m, so PQ and PR should be perpendicular to lines I and m respectively] OP = OP [comon side] ∴ΔOQP≅ΔORP [by RHS congruence rule] ⇒∠POQ=∠POR [by CPCT] Hence, n is the bisector of ∠QOR.

Question 15
Two lines I 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 I and m. Prove that n is the bisector of the angle formed by I and m.