In the given figure A, B, C andD, E, F are two sets of collinearpoints

1.	In the given figure A, B, C andD, E, F are two sets of collinearpoints. Prove that AD \|CF.2
Answer» In this figure, ABED is a cyclic quadrilateral. ∠DAB = ∠ BED ( as opposite angles of a cyclic quadrilateral are supplementary) ...(1) BCFE is also a cyclic Quad. ∴ ∠BCF = ∠ BEF(as opposite angles of a cyclic quadrilateral are supplementary)…(2) when we add these angles ∠DAB + ∠ BED + ∠BCF + ∠ BEF = 360°…(1) ∠BED + ∠ BEF = 180° ⇒ ∠ DAB + ∠ BCF = 180° But they are co-interior angles and if co-interior angles sum up to 180° they are parallel. So AD parallel to CF.

In the given figure A, B, C andD, E, F are two sets of collinearpoints. Prove that AD |CF.2

Answer»

In this figure, ABED is a cyclic quadrilateral.

∠DAB = ∠ BED ( as opposite angles of a cyclic quadrilateral are supplementary) ...(1)

BCFE is also a cyclic Quad.

∴ ∠BCF = ∠ BEF(as opposite angles of a cyclic quadrilateral are supplementary)…(2)

when we add these angles

∠DAB + ∠ BED + ∠BCF + ∠ BEF = 360°…(1)

∠BED + ∠ BEF = 180°

⇒ ∠ DAB + ∠ BCF = 180°

But they are co-interior angles and if co-interior angles sum up to 180° they are parallel.

So AD parallel to CF.