Mathematics
In the given figure, ABCD is a cyclic quadrilateral. AF is drawn parallel to CB and DA is produced to point E. If ∠ADC = 92°, ∠FAE = 20°; determine ∠BCD. Given reason in support of your answer.
Circles
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Answer
Given,
In cyclic quad. ABCD
AF || CB and DA is produced to E such that ∠ADC = 92° and ∠FAE = 20°.
From figure,
⇒ ∠B + ∠D = 180° [As sum of opposite angles in a cyclic quadrilateral = 180°]
⇒ ∠B + 92° = 180°
⇒ ∠B = 180° - 92° = 88°
As AF || CB,
∠FAB = ∠B = 88° [Alternate angles are equal]
But, ∠FAE = 20° [Given]
From figure,
∠BAE = ∠BAF + ∠FAE = 88° + 20° = 108°.
∠BAD = 180° - ∠BAE = 180° - 108° = 72°.
∠BCD + ∠BAD = 180° [As sum of opposite angles in a cyclic quadrilateral = 180°]
⇒ ∠BCD + 72° = 180°
⇒ ∠BCD = 180° - 72° = 108°.
Hence, ∠BCD = 108°.
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