Question

FOB and DOA are straight lines. Find the measure of $\angle AOC$.

• A. ${112}^{\circ }$
• B. ${66}^{\circ }$
• C. ${122}^{\circ }$
• D. ${46}^{\circ }$

Answer 1

The correct option is A ${112}^{\circ }$

$\angle EOC={90}^{\circ }$ (Given)
$\angle EOD+\angle DOC={90}^{\circ }$ (Complementary angles)
$\Rightarrow x^{\circ }+(4x-20{)}^{\circ }={90}^{\circ }$
$\Rightarrow 5x-20=90$
$\Rightarrow 5x=110$
$\Rightarrow x=22$
$\angle EOD=x^{\circ }={22}^{\circ }$
$\angle DOC=(4x-20{)}^{\circ }=(4\times 22-20{)}^{\circ }={68}^{\circ }$
Now, FOB and DOA are straight lines.
So,
$\angle AOB=\angle FOD$ (Vertical angles)
$\Rightarrow \angle AOB=\angle FOE+\angle EOD$
$\Rightarrow \angle AOB={24}^{\circ }+{22}^{\circ }$
$\Rightarrow \angle AOB={46}^{\circ }$
$\angle FOE+\angle EOC+\angle BOC={180}^{\circ }$ (Supplementary angles)
${24}^{\circ }+{90}^{\circ }+\angle BOC={180}^{\circ }$
$\Rightarrow \angle BOC={180}^{\circ }-{114}^{\circ }={66}^{\circ }$
Now,
$\angle AOC=\angle AOB+\angle BOC$
$\angle AOC={46}^{\circ }+{66}^{\circ }$
$\angle AOC={112}^{\circ }$