Question

In the figure $AC$ is the diameter of the circle with centre $O$. If $CD\parallel BE,\angle ACE={10}^{\circ },\angle AOB={80}^{\circ }$, find :
$i)\angle BEC$
$ii.)\angle BCD$
$iii.)\angle CED$

Answer 1

$\begin{array}{c}\angle BOC={180}^{\circ }-{80}^{\circ }={100}^{\circ }\\ \Rightarrow \angle BEC={50}^{\circ }\end{array}$ Angle subtended by arc at centre is double as that of at the circumference.$\begin{array}{c}\left[alternateangles\right]\\ \angle BOC+\angle OCB+\angle OBC\\ \left[\angle OCB=\angle OBC\right]\\ \because OC=OB\\ \Rightarrow \angle OCB={40}^{\circ }\\ \Rightarrow \angle ECD={50}^{\circ }\\ \angle OCB={40}^{\circ }\\ \angle BCD={50}^{\circ }+{40}^{\circ }+{10}^{\circ }\\ \angle BCD={100}^{\circ }\\ \angle BCD+\angle DEB={180}^{\circ }\\ \angle DEB={80}^{\circ }\\ \angle DEC={30}^{\circ }\\ \angle CED={30}^{\circ }\end{array}$