The conveise Ur TIcuIUlTheorem 10.10 : If a line segmeni joining two p

1.	The conveise Ur TIcuIUlTheorem 10.10 : If a line segmeni joining two points subtends equal angles attwo other points lying on the same side of the line containing the line segment,ihe four points lie on a circle (i.e. they are concyclic)os
Answer» Given:AB is a line segment and C and D are two points lying on the same side of AB such that ∠ACB = ∠ADB. To prove:A, B, C and D are concyclic. Proof: If possible, suppose D does not lie on this circle. Let D´ be the point which lies on the circle. Since, C and D´ are two point on the circle lying on the same side of AB. ∴ ∠ACB = ∠AD´B (Angles in the same segment are equal) ⇒ ∠ADB = ∠AD´B (∠ACB = ∠ADB) ∴ An exterior of ΔDBD´ is equal to the interior opposite angle. But, an exterior angle of a triangle canneverbe equal to its interior opposite angle. ∴ ∠ADB = ∠AD´B ⇒ D coincides with D´. ⇒ D lies on the circle passing through the points A, B and C. Hence, the points A, B, C and D are concyclic.

The conveise Ur TIcuIUlTheorem 10.10 : If a line segmeni joining two points subtends equal angles attwo other points lying on the same side of the line containing the line segment,ihe four points lie on a circle (i.e. they are concyclic)os

Answer»

Given:AB is a line segment and C and D are two points lying on the same side of AB such that ∠ACB = ∠ADB.

To prove:A, B, C and D are concyclic.

Proof:

If possible, suppose D does not lie on this circle. Let D´ be the point which lies on the circle.

Since, C and D´ are two point on the circle lying on the same side of AB.

∴ ∠ACB = ∠AD´B (Angles in the same segment are equal)

⇒ ∠ADB = ∠AD´B (∠ACB = ∠ADB)

∴ An exterior of ΔDBD´ is equal to the interior opposite angle. But, an exterior angle of a triangle canneverbe equal to its interior opposite angle.

∴ ∠ADB = ∠AD´B

⇒ D coincides with D´.

⇒ D lies on the circle passing through the points A, B and C.

Hence, the points A, B, C and D are concyclic.