Question

ABCD is a cyclic quadrilateral. Sides AB and DC produced meet at point E; whereas sides BC and AD produced meet at point F.  If ∠DCF : ∠F : ∠E = 3 : 5 : 4, find the angles of the cyclic quadrilateral ABCD.

Answer 1

Given – In a circle, ABCD is a cyclic quadrilateral AB and DC are produced to meet at E and BC and AD are produced to meet at F.

∠DCF : ∠F : ∠E = 3 : 5 : 4

Let ∠DCF = 3x, ∠F = 5x, ∠E = 4x

Now, we have to find, ∠A, ∠B, ∠C and ∠D

In cyclic quad. ABCD, BC is produced.

∴ ∠A = ∠DCF = 3x

In ΔCDF,

Ext ∠CDA = ∠DCF + ∠F = 3x + 5x = 8x

In ΔBCE,

Ext ∠ABC = ∠BCE + ∠E   ...[∠BCE = ∠DCF, Vertically opposite angles]

= ∠DCF + ∠E

= 3x + 4x

= 7x

Now, in cyclic quad ABCD,

Since, ∠B + ∠D = 180°  ...[Since sum of opposite of a cyclic quadrilateral are supplementary]

`=>` 7x + 8x = 180°

`=>` 15x = 180°

`=> x = (180^circ)/15 = 12^circ`

∠A = 3x = 3 × 12° = 36°

∠B = 7x = 7 × 12° = 84°

∠C = 180° – ∠A = 180° – 36° = 144°

∠D = 8x = 8 × 12° = 96°