In the figure (ii) given below, AB is parallel to DC, ∠BCE = 80° and ∠BAC = 25°. Find :

(i) ∠CAD

(ii) ∠CBD

(iii) ∠ADC

(i) ABCD is a cyclic quadrilateral.

Since exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.

∴ ∠BCE = ∠DAB

⇒ ∠BCE = ∠CAD + ∠BAC
⇒ 80° = ∠CAD + 25°
⇒ ∠CAD = 80° - 25°
⇒ ∠CAD = 55°.

Hence, the value of ∠CAD = 55°.

(ii) From figure,

∠CBD = ∠CAD (∵ angles in same segment are equal.)

∠CBD = 55°.

Hence, the value of ∠CBD = 55°.

(iii) ∠BAC = ∠BDC (∵ angles in same segment are equal.)

∠BDC = 25°.

Now AB || DC and BD is the transversal

∠ABD = ∠BDC (∵ ∵ alternate angles are equal.)

⇒ ∠ABD = 25°

From figure,

∠ABC = ∠ABD + ∠CBD = 25° + 55° = 80°.

Since, sum of opposite angles of a cyclic quadrilateral is 180°.

⇒ ∠ABC + ∠ADC = 180°
⇒ 80° + ∠ADC = 180°
⇒ ∠ADC = 180° - 80°
⇒ ∠ADC = 100°.

Hence, the value of ∠ADC = 100°.