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

[Hint: Use proof by contradiction]


Let us draw a line m. Let A be a point.


Let us draw two intersecting lines l and p through the point A, such that they are perpendicular to the line m at points P and Q.


P = 90°, and Q = 90° - - - - (i)


We have to prove that through a given point, we can draw only one perpendicular to a given line.


To prove that A = 0°


Now, we can see in ΔPAQ,


P + A + Q = 180° (Since the sum of the angles of a triangle is equal to 180°)


A + 90° + 90° = 180° [From equation (i)]


A + 180° = 180°


A = 0°


This means that A doesn’t exist, i.e., the lines l and p coincide with each other.


Therefore, it is true that through a given point, we can draw only one perpendicular to a given line.


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