Mathematics
ABCD is a cyclic quadrilateral in which BC = CD and EF is a tangent at A. ∠CBD = 43° and ∠ADB = 62°. Find :
(a) ∠ADC
(b) ∠ABD
(c) ∠FAD
Circles
ICSE Sp 2024
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Answer
(a) In △ BDC,
⇒ BC = CD (Equal sides)
⇒ ∠BDC = ∠DBC = 43° (Angles opposite to equal sides are equal)
From figure,
⇒ ∠ADC = ∠ADB + ∠BDC = 62° + 43° = 105°.
Hence, ∠ADC = 105°.
(b) We know that,
Opposite angles of a cyclic quadrilateral are supplementary.
⇒ ∠ABC + ∠ADC = 180°
⇒ ∠ABC + 105° = 180°
⇒ ∠ABC = 180° - 105° = 75°.
From figure,
⇒ ∠ABD = ∠ABC - ∠DBC = 75° - 43° = 32°.
Hence, ∠ABD = 32°.
(c) From figure,
⇒ ∠FAD = ∠ABD = 32°. (Angles in alternate segment are equal)
Hence, ∠FAD = 32°.
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