1. THEOREM CONVERSE OF Alternate segment Theorem Statement:- “If a line is drawn through an end point of a chord of a circle so that the angle formed with the chord is equal to the angle substended by the chord in the alternate segment, then the line is a tangent to the circle”. C B o x1 X Y A y1

2. Given:- circle with centre “O” XAY is a line meeting chord AB at A
Given:- circle with centre “O” XAY is a line meeting chord AB at A. and C is the point in the other segment such that BAY = ACB. C B o x1 X Y A y1

3. R.T.P:- XAY is tangent to the circle Construction:- NILB o x1 X Y A y1

4. Proof:- suppose XAY is not tangent to the circle
Proof:- suppose XAY is not tangent to the circle. Let X‘AY’ is tangent to the circle Let X‘AY’ is tangent to the circle ACB = BAY (1) C B o x1 X Y A y1

5. But ACB = BAY (given) (2) From (1) & (2) BAY = BAY which is possible iff XAY coincides with XAY XAY is tangent to the circle at A. C B o x1 X Y A y1

6. Conclusion:- “If a line is drawn through and end point of a chord of a circle so that the angle formed with the chord is equal to the angle substended by the chord in the alternate segment, then the line is a tangent to the circle”. C B o x1 X Y A y1