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Question
Prove that through a given point, we can draw only one perpendicular to a given line.
[Hint: Use proof by contradiction].
Solution
Given Consider a line l and a point P.
Construction: Draw two intersecting lines passing through the point P and which is perpendicular to l.
To prove: Only one perpendicular line can be drawn through a given point i.e., to prove ∠P = 0°.
Proof: In ΔAPB, ∠A + ∠P + ∠B = 180° ...[By angle sum property of a triangle is 180°]
⇒ 90 + ∠P + 90° = 180°
⇒ ∠P = 180° – 180°
∴ ∠P = 0°
So, lines n and m coincide.
Hence, only one perpendicular line can be drawn through a given point.
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