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ABCD is a cyclic quadrilateral in which O is the center of the circle and one of the diagonal of the cyclic quadrilateral passes through the center of the circle and ∠DCA = 30°. Find the measure of ∠CAD.

ABCD is a cyclic quadrilateral in which O is the center of the circle and one of the diagonal of the cyclic quadrilateral passes through the center of the circle and ∠DCA = 30°. Find the measure of ∠CAD. Correct Answer 60° 

Given:

ABCD is a cyclic quadrilateral

'O' is the center of a circle

∠DCA = 30° 

Concept used:

In the cyclic quadrilateral, the sum of the opposite angle is 180° 

And angle made in a semi-circle is 90° 

The sum of all angle inside a triangle = 180° 

Calculation:

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Let us assume AC be diagonal that passes through the center

From the above figure, we can see that AC is the diagonal and also AC is the diameter of the circle and then ∠ABC = ∠ADC = 90°  

In ΔADC,

∠CAD + ∠ADC + ∠DCA = 180°

⇒ ∠CAD + 90° + 30° = 180°

⇒ ∠CAD + 120° = 180° 

⇒ ∠CAD = 180° - 120° 

⇒ ∠CAD = 60° 

∴ The measure of ∠CAD is 60°. 

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