Prove the theorem of alternate segment

Alternate Segment:

If a chord is drawn through the point of contact of a tangent to a circle, then the angles which this chord makes with the given tangent are equal respectively to the angles formed in the corresponding alternate segments.

Given : is the tangent to the circle with centre O. AB is the chord drawn at the point of contact A.

C and D are two points on the circumference such that they lie on either side of the chord.

To Prove: (i) and (ii)

Construction: Draw the diameter AOE and draw EB.

Proof: In the figure, ( radius is perpendicular to the tangent at the point of contact)

… (1)

In DEAB

(Angle in a semi-circle)

… (2)

From (1) and (2)

But, (Angles in the same segment)

Now, consider cyclic quadrilateral ACBD.

(Opposite angles) ….(4)

And (Linear pair)

But

… (5)

From (4) and (5)

Hence proved.