In the given figure, find whether the points A, B, C, D are concyclic when:

(i) x = 70

(ii) x = 80

From the figure,

∠ABC + ∠CBE = 180° [Linear pair]

∠ABC + 110° = 180°

∠ABC = 70°

For the points A, B, C, D to be concyclic, ∠ABC + ∠ADC must be equal to 180°.

(i) When x = 70

x° + ∠ADC = 180° [Linear pair]

∠ADC = 180° − x° = 180° - 70° = 110°.

∠ABC + ∠ADC = 70° + 110° = 180°

Since the sum of opposite angles is 180°, the points A, B, C, D are concyclic when x = 70.

Hence, yes the points A, B, C, D are concyclic when x = 70.

(ii) When x = 80

x° + ∠ADC = 180° [Linear pair]

∠ADC = 180° − 80° = 100°

Check for con-cyclicity,

∠ABC + ∠ADC = 70° + 100° = 170°

Since the sum of opposite angles is not 180°, the points A, B, C, D are not concyclic when x = 80.

Hence, no the points A, B, C, D are not concyclic when x = 80.