Alternate segment theorem


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A line that touches a circle at only one point is known as a tangent to the circle.
The common point to the tangent and the circle is known as the point of contact.
Alternate segment theorem

If a line touches a circle and from the point of contact a chord is drawn, then the angles that this chord makes with the given line are equal to the angles formed in the corresponding alternate segments, respectively. 
Given: Let AB be a chord of a circle with centre O. PQ be a tangent to the circle at A. 
Let E and F be any two points on the circle such that they are in alternate segments R2 and R1, respectively. 
To prove: (i) m BAQ = m AEB     
              (ii)  m BAP = m AFB
  AEB is an inscribed angle in arc AEB, while arc AFB is the intercepted arc.
  m AEB = (1/2)m( arc AFB) (By inscribed angle theorem) ….. (1)     
 PQ is tangent at A and line AB is a secant. 
 BAQ intercepts arc AFB
  m BAQ = (1/2)m( arc AFB) (By tangent secant theorem) ….. (2) 
From (1) and (2)
 m AEB =  m BAQ
Similarly,   m AFB =  m BAP.
Converse of Alternate Segment Theorem 
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 subtended by the chord in the alternate segment, then the line is a tangent to the circle. 
Given: AB is a chord of a circle with centre O. 
Line PAQ is drawn through A such that m BAQ = m ACB , where C is a point on the circumference. 

To prove : PAQ is a tangent to the circle at point A.
Let us suppose that PAQ is not a tangent. 
Draw a tangent P'AQ'  to the circle at A.
 m BAQ' = m ACB (By alternate segment theorem)
But m BAQ = m ACB    (Given)
  m BAQ = m BAQ'
Unless ray AQ'   coincides with  AQ, this is impossible. 
Therefore, P'AQ'  coincides with PAQ.
Or PAQ is a tangent to the circle at A.

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