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​Q11. In the given figure, A, B, C and D, E, F are two sets of collinear points. Prove that : AD || CF.

         

Dear Student,
Here is the solution of your asked query:

Q.111. Given: Two circles intersect at points B and E. A, B and C and D, E and F are collinear points.

To prove: AD CF

Proof:

Since, ABED is a cyclic quadrilateral.  (points A, B, E and D lie on the circle)

So, ∠1 + ∠2 = 180°            (opposite angles of a cyclic quadilateral are supplementary)

Also, ∠2 + ∠3 = 180°          (linear pair)

Thus, ∠1 = ∠3               ...(i) 

Now, in cyclic quadilateral BCFE,

∠3 + ∠4 = 180°

∠1 + ∠4 = 180°              (using (i))

Now consider lines AD and BF and transversal AB,

Since, the sum of angles on the same side of transversal is 180°.

Hence, AD CF.

Hence proved.

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