Theorem 4. If a line is drawn through an end point of a chord of a circle so that the angle formed by it with the chord is equal to the angle subtend by chord in the alternate segment, then the line is a tangent to the circle.
Given:- A chord AB of a circle and a line PAQ. Such that where c is any point in the alternate segment ACB.
To Prove:- PAQ is a tangent to the circle.
Construction:- Let PAQ is not a tangent then let us draw P' AQ' another tangent at A.
Proof: - AS P ’AQ’ is tangent at A and AB is any chord
[theo.3]
But (given)
Hence AQ' and AQ are the same line i.e. P' AQ' and PAQ are the same line.
Hence PAQ is a tangent to the circle at A.
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