Question

Question 4
Prove that through a given point, we can draw only one perpendicular to a given line.

Answer 1

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 $\angle P=0^{\circ }$.
Proof:
In $\Delta APB,\angle A+\angle P+\angle B={180}^{\circ }$
[by angle sum property of a triangle]
$\Rightarrow {90}^{\circ }+\angle P+{90}^{\circ }={180}^{\circ }$
$\Rightarrow \angle P={180}^{\circ }-{180}^{\circ }$
$\therefore \angle P=0^{\circ }$
So, lines n and m coincide.
Hence, only one perpendicular line can be drawn through a give point.