Discussion
In the following figure, ABCD is a cyclic quadrilateral in which AC and BD are its diagonals. If ∠ DBC = 55° and ∠ BAC = 45°, then the value of ∠ BCD will be :
Answer: C
The value of ∠ BCD is 80°.
Angles in the same segment of a circle are equal. ∠ BAC and ∠ BDC both stand on the chord BC, so ∠ BDC = ∠ BAC = 45°. In triangle BCD the angles add up to 180°:
\[ \begin{aligned} \angle BCD &= 180^\circ - (55^\circ + 45^\circ) \\ &= 80^\circ \end{aligned} \]
90° would assume BD is a diameter, which nothing in the question says. 65° and 70° do not follow from the given angles.
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