In the figure (i) given below, M, A, B, N are points on a circle having center O. AN and MB cut at Y. If ∠NYB = 50° and ∠YNB = 20°, find ∠MAN and the reflex angle MON.

Considering △YBN,

Since sum of angles in a triangle is equal to 180°.

⇒ ∠NYB + ∠YNB + ∠YBN = 180°.
⇒ 50° + 20° + ∠YBN = 180°
⇒ 70° + ∠YBN = 180°
⇒ ∠YBN = 180° - 70°
⇒ ∠YBN = 110°.

From figure,

∠MBN = ∠YBN = 110°.

Considering △MAN and △MBN,

∠MAN = ∠MBN = 110° (∵ angles in same segment are equal.)

MN subtends Reflex ∠MON at center and ∠MAN at point A of circle.

⇒ Reflex ∠MON = 2 ∠MAN = 2 × 110° = 220°.

Hence, ∠MAN = 110° and Reflex ∠MON = 220°.