In the given figure, AE is the diameter of the circle. Find \(\angle ABC+\angle CDE\):
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 270°
Join AC and CE. Two facts decide the question: the opposite angles of a cyclic quadrilateral add up to 180°, and the angle in a semicircle is 90°.
ABCE is a cyclic quadrilateral: \(\angle ABC+\angle AEC=180^\circ\) (1)
ACDE is a cyclic quadrilateral: \(\angle CDE+\angle CAE=180^\circ\) (2)
Adding (1) and (2): \(\angle ABC+\angle CDE=360^\circ-(\angle AEC+\angle CAE)\)
AE is a diameter, so \(\angle ACE=90^\circ\), and in triangle ACE, \(\angle AEC+\angle CAE=180^\circ-90^\circ=90^\circ\)
\(\angle ABC+\angle CDE=360^\circ-90^\circ=270^\circ\)