Mathematics
ABCD is a cyclic quadrilateral such that AC is a diameter of the circle. If ∠BAC = 58° and ∠DAC = 65°, then ∠BCD is equal to :
57°
123°
90°
60°
Answer
∠ADC = 90° and ∠ADC = 90° [Angle in semicircle is a right angle]
In △ABC,
∠BAC = 58°
∠ABC = 90°
By angle sum property of triangle,
∠BCA + ∠BAC + ∠ABC = 180°
∠BCA = 180° - (∠BAC + ∠ABC)
∠BCA = 180° - (90° + 58°) = 32°
In △ABC :
∠DAC = 65° and ∠ADC = 90°
By angle sum property of triangle,
∠DCA + ∠DAC + ∠ADC = 180°
∠DCA = 180° - (∠DAC + ∠ADC)
∠DCA = 180° - (90° + 65°) = 25°
From figure,
∠BCD = ∠DCA + ∠BCA = 32° + 25° = 57°.
Hence, option 1 is the correct option.
Related Questions
In the given figure, O is the centre of the circle.
If ∠PQR = 35°, then ∠POR is equal to :17.5°
35°
60°
70°
In the given figure, O is the centre of the circle and ∠BDC = 36°. The measure of ∠ACB is :
36°
54°
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The angle formed in a minor segment of a circle is :
an acute angle
an obtuse angle
a right angle
either an acute or an obtuse angle
The angle formed in a major segment of a circle is :
an acute angle
an obtuse angle
a right angle
either an acute or an obtuse angle