Question

Find the measure of $\angle CAD\,\,and\,\,\angle BCD$

Answer 1

Hint: We will use angle in the segment theorem, where the base is the same between two angles. It will help to calculate $\angle BCD$ and $\angle CAD$.


Complete step by step solution:
In circle, $\angle CAD$ and$ \angle CBD$ have same arc $DC$, so we use theorem angle in the same segment

Therefore,
$\angle CAD = \angle CBD$
As we know that $\angle CBD = 40^\circ $, then $\angle CAD = 40^\circ $
Now similarly, $\angle CBD$ and $ \angle CDB$ have same arc $BC$, we use theorem angle in the same segment,
Therefore,
$\angle CDB = \angle CAB$
As we know that$\angle CAB = 60^\circ $then,
$\angle CDB = 60^\circ $
Now, in$\Delta BCD,$
$\angle CBD + \angle CAB + \angle DCB = 180^\circ $(Angle sum property)
$40^\circ + 60^\circ + \angle DCB = 180^\circ $
$
  100^\circ + \angle BCD = 180^\circ \\
  \angle BCD = 180^\circ - 100^\circ \\
  \angle BCD = 80^\circ \\
 $
Thus, $\angle BCD = 80^\circ $


Note: The angles at the circumference subtended by the same arc are equal. It means two angles should have one common chord, if you are taking two chords then you will not be able to find the angle between them.