Question

In the given figures, O is the centre of the circles. Find the respective values of x.

• A. 60°
• B. 115°
• C. 40°

Answer 1

In the first image,we see that AB=CD.
We know that equal chords subtend equal angles at the centre.
$⟹\angle AOC=\angle BOD={60}^{\circ }$

In the second image,AB=CD and $\angle AOB={40}^{\circ }.$
Since equal chords subtend equal angles at the centre, we have
$\angle AOB=\angle COD={40}^{\circ }.$
Note that $△$ AOB and $△$COD are isosceles.
Thus, using angle sum property in $△COD,$ we have
$\angle COD+\angle OCD+\angle ODC={180}^{\circ }.$
$⟹40+x+x={180}^{\circ }$
$⟹x={70}^{\circ }$

In the third image,
$\angle OCA+\angle OCB=180°$
$\angle OCB=105°$
Now, using angle sum property in $△OCB,$ we have
${30}^{\circ }+{105}^{\circ }+\angle BOC={180}^{\circ }⟹\angle BOC={40}^{\circ }.$
Now, in $\Delta ACO$
$\angle OAC+\angle OCA+\angle AOC=180°$
$30°+75°+\angle AOC=180°$
$\angle AOC={75}^{\circ }.$
Thus, $x=\angle AOC+\angle BOC=75°+40°=115°$