(a) In the figure (i) given below, M, A, B, N are points on a circle having centre O. AN and MB cut at Y. If ∠NYB = 50° and ∠YNB = 20°, find ∠MAN and the reflex angle MON. (b) In the figure (ii) given below, O is the centre of the circle. If ∠AOB = 140° and ∠OAC = 50°, find (i) ∠ACB (ii) ∠OBC (iii) ∠OAB (iv) ∠CBA
(a) In the figure (i) given below, M, A, B, N are points on a circle having centre O. AN and MB cut at Y. If ∠NYB = 50° and ∠YNB = 20°, find ∠MAN and the reflex angle MON. (b) In the figure (ii) given below, O is the centre of the circle. If ∠AOB = 140° and ∠OAC = 50°, find (i) ∠ACB (ii) ∠OBC (iii) ∠OAB (iv) ∠CBA

Solution

(a) ∠NYB = 50°, ∠YNB = 20°.

In
∆YNB,

NYB + ∠YNB + YBN = 180o

50+ 20+ ∠YBN = 180o

YBN + 70= 180o

YBN = 180– 70= 110o

But MAN = YBN

(Angles in the same segment)

MAN = 110o

Major arc MN subtend reflex MON at the

Centre and MAN at the remaining part of

the choice.

Reflex MAN at the remaining part of the circle

Reflex MON = 2 MAN = 2 × 110=220o

(b) (i)
AOB + reflex AOB = 360o

(Angles at the point)

140+ reflex AOB = 360o

Reflex AOB = 360– 140= 220o

Now major arc AB subtends AOB OBC = 360o

50+ 110+ 140OBC = 3600

300+ ∠OBC = 3600

∠300o + ∠OBC = 3600

∠OBC = 360o – 300o

∠OBC = 60o

(ii) In Quadrilateral .OACB

∠OAC + ∠ACB + ∠AOB + ∠OBC = 360o

50+ 110+ 140o + ∠OBC = 360o

300o + ∠OBC = 360o

∠OBC = 360o – 300o

∠OBC =60o

(iii) in ∆OAB,

OA = OB

(Radii of the same circle)

∠OAB + ∠OBA = 180o

2 ∠OAB = 180o – 140= 40o

∠OAB = 40o/2 = 200

But ∠OBC = 60o

∠CBA = ∠OBC – ∠OBA

= 60o – 20= 40o