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

Question

Philoid · Accepted Answer

Given: AB is a line and P is any point outside AB. PM ⊥ AB and PN is another line drawn from P to AB.
To prove: We can draw only one perpendicular from a given point P to a given line AB.
Proof: Let PM and PN be two perpendiculars drawn from a point P to the line AB.
Then ∠ 1 = ∠ 2 = 90°
But these two angles form a pair of corresponding angles.
∴ PM || PN
But two parallel lines never intersect.
So our assumption is wrong.
∴ It is proved that we can draw only one perpendicular to a given line through a given point.