Question : $ABCD$ is a cyclic quadrilateral of which $AB$ is the diameter. Diagonals $AC$ and $BD$ intersect at $E$. If $\angle DBC=35^\circ$, then $\angle AED$ measures _________
Option 1: 35$^\circ$
Option 2: 45$^\circ$
Option 3: 55$^\circ$
Option 4: 90$^\circ$
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Correct Answer: 55$^\circ$
Solution :
Given, a cyclic quadrilateral $ABCD$ with $AB$ as a diameter
And, $\angle DBC=35^\circ$
We know, $\angle ACB=90^\circ$ (angle in semicircle)
In $\triangle BEC$,
$\angle ECB+\angle EBC+\angle BEC=180^\circ$
or, $\angle BEC=180^\circ-90^\circ-35^\circ$
So, $\angle BEC=\angle AED = 55^\circ$ (vertically opposite angle)
Hence, the correct answer is 55$^\circ$.
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