Question

In the given figure, DC is diameter then $x$ = ___.

• A. ${60}^{\circ }$
• B. ${70}^{\circ }$
• C. ${80}^{\circ }$
• D. ${100}^{\circ }$

Answer 1

The correct option is D ${100}^{\circ }$

As DC is diameter,
We know that angle in a semicircle is a right angle.
$⟹\angle DBC={90}^{\circ }$
$⟹\angle OBC=\angle DBC-\angle DBO={90}^{\circ }-{40}^{\circ }={50}^{\circ }$
Now, in $\Delta BOC$

OB = OC ( radii of same circle)

$\therefore$ $\angle OCB=\angle OBC$ (sides opposite to equal angles are equal)

Therefore, $\angle OCB={50}^o$

Now, in $\Delta BOC$

$\angle OBC+\angle OCB+\angle BOC={180}^0$

$\therefore {50}^0+{50}^0+\angle BOC={180}^0$

or $\angle BOC={180}^0-{100}^0$

$={80}^0$

$\therefore \angle BOC={80}^0$

Now, $\angle AOC+\angle BOC={180}^0\left(linearpair\right)$

$\therefore x+{80}^0={180}^0$

Hence $x={180}^0-{80}^0$

$={100}^0$