Mathematics
In the given figure, ∠ADC = 130° and BC = BE. Find ∠CBE if AB ⊥ CE.
Circles
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Answer
Since, ABCD is a cyclic quadrilateral and sum of opposite angles in a cyclic quadrilateral = 180°.
∴ ∠ADC + ∠ABC = 180°
⇒ 130° + ∠ABC = 180°
⇒ ∠ABC = 50°.
From figure,
⇒ ∠FBC = ∠ABC = 50°.
In △FBC,
By angle sum property of triangle,
⇒ ∠FBC + ∠BCF + ∠CFB = 180°
⇒ 50° + ∠BCF + 90° = 180°
⇒ ∠BCF = 180° - 90° - 50° = 40°.
Given,
BC = BE
In △BCE,
⇒ ∠BEC = ∠BCE = 40°. (As angles opposite to equal sides are equal)
In △FBE,
By angle sum property of triangle,
⇒ ∠BEF + ∠FBE + ∠EFB = 180°
⇒ 40° + ∠FBE + 90° = 180°
⇒ ∠FBE = 180° - 90° - 40° = 50°.
From figure,
∠CBE = ∠FBC + ∠FBE = 50° + 50° = 100°.
Hence, ∠CBE = 100°.
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