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  • Prove that through a given point, we can draw only one perpendicular to a given line. [Hint: Use proof by contradiction].

Prove that through a given point, we can draw only one perpendicular to a given line. [Hint: Use proof by contradiction].

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Solution:

Given: Consider a line R and a point P

Construction:
Draw two lines (m and n) passing through P which are perpendicular to line R.
To prove: Only one perpendicular line can be drawn through a point P
Proof: In \triangle APB
\angle A + \angle P + \angle B = 180^{\circ}   {angle sum property}

90^{\circ} + \angle P + 90^{\circ} = 180^{\circ}
\angle P = 180 - 180^{\circ}
\angle P = 0^{\circ}
So lines n and m will coincide
Therefore we can draw only one perpendicular to a given line.

Hence proved

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