Question

prove that the tangent at the end of a diameter of a circle are parallel

Answer 1

Given: A circle with center O And diameter AB Let PQ be the tangent at point A & RS be the tangent at point B

To prove: PQ ∥ RS

Proof: Since PQ is a tangent at point A
OA ⊥ PQ
∠OAP = 90°...............(1)
Similarly, RS is a tangent at pt B
OB⊥RS
∠OBS=90°.................(2)
from (1)and(2)
∠OAP =∠OBS=90°
so,
∠BAP=∠ABS
so, the two lines PQ and RS are transversals
and both the angles are alternate interior angles
so, the the tangents are parallel

figure is here https://hi-static.z-dn.net/files/d70/32f7aa75797920c5ebfe7f4f88a281f1.jpg

Answer 2

Here Theorem 1 is: The tangent at any point of a circle is perpendicular to the radius through the point of contact