Question

Prove that any four vertices of a regular pentagon are concylic (lie on the same circle).

Prove that any four vertices of a regular pentagon lie on a circle.

Answer 1

ABCDE is a regular pentagon.

∴ ∠BAE = ∠ABC = ∠BCD = ∠CDE = ∠DEA = ((5 - 2)/5) xx 180^circ = 180^circ`

In ΔAED,

AE = ED  ...(Sides of regular pentagon ABCDE)

∴ ∠EAD = ∠EDA

In ΔAED,

∠AED + ∠EAD + ∠EDA = 180°

⇒ 108° + ∠EAD + ∠EAD = 180°

⇒ 2∠EAD = 180° − 108° = 72°

⇒ ∠EAD = 36°

∴ ∠EDA = 36°

 ∠BAD = ∠BAE − ∠EAD = 108° − 36° = 72°

In quadrilateral ABCD,

∠BAD + ∠BCD = 108° + 72° = 180°

∴ ABCD is a cyclic quadrilateral.