In a regular pentagon ABCDE, inscribed in a circle; find the ratio between angle EDA and angle ADC.

Regular pentagon ABCDE inscribed in a circle is shown in the figure below:

We know that,

Angle at the centre is twice the angle at remaining circumference.

∴ ∠AOE = 2∠ADE

⇒ ∠ADE = $\dfrac{1}{2}$∠AOE

As ∠AOE is subtended by AE which is a side of a regular pentagon inscribed in a circle,

∴ ∠AOE = $\dfrac{360°}{5}$ = 72°

⇒ ∠ADE = $\dfrac{1}{2}$∠AOE = $\dfrac{1}{2} \times 72°$ = 36°.

We know that,

Each side of interior angle of a regular pentagon = 108°.

From figure,

⇒ ∠ADC = ∠EDC - ∠ADE = 108° - 36° = 72°.

∴ ∠ADE : ∠ADC = 36° : 72° = 1 : 2.

Hence, the ratio between angle EDA and angle ADC = 1 : 2.