Question

In Fig. 16.189, ABCD is a cyclic quadrilateral in which AC and BD are its diagonals. If ∠DBC=55° and ∠BAC=45°, find ∠BCD.

Answer 1

Given :  ABCD is a cyclic quadrilateral in which AC and BD are its diagonals and ∠DBC = 55° and ∠BAC = 45°.

 

To Find : ∠BCD

 

Proof :  

Since angles in the same segment of a circle are equal :

∴ ∠CAD = ∠DBC = 55°

∴∠DAB = ∠CAD + ∠BAC

∠DAB = 55°  +  45°

∠DAB = 100°

Since, ABCD is a cyclic quadrilateral, and Sum of Opposite pair of angles in a  cyclic quadrilateral is 180° :  

∠DAB + ∠BCD = 180°

∴∠BCD = 180° - ∠DAB

∠BCD = 180o - 100°

∠BCD = 80°

Hence the value of ∠BCD is 80°.

 

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In Fig. 16.188, ABCD is a cyclic quadrilateral. Find the value of x.

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Answer 2

Given :  

ABCD is a cyclic quadrilateral in which AC and BD are its diagonals and ∠DBC = 55° and ∠BAC = 45°.

 

To Find :

∠BCD

 

Proof :  

Since angles in the same segment of a circle are equal :

∴ ∠CAD = ∠DBC = 55°

∴∠DAB = ∠CAD + ∠BAC

∠DAB = 55°  +  45°

∠DAB = 100°

Since, ABCD is a cyclic quadrilateral, and Sum of Opposite pair of angles in a  cyclic quadrilateral is 180° :  

∠DAB + ∠BCD = 180°

∴∠BCD = 180° - ∠DAB

∠BCD = 180o - 100°

∠BCD = 80°

Hence the value of ∠BCD is 80°.

 

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