Question

In the given figure, $\angle$ABO = 45${}^{\circ }$ and $\angle$BAO = 75${}^{\circ }$. If $\angle$OCD = 70${}^{\circ }$, what is the value of $\angle$ODC?

• A. 45${}^{\circ }$
• B. 50${}^{\circ }$
• C. 60${}^{\circ }$
• D. 70${}^{\circ }$

Answer 1

The correct option is B 50${}^{\circ }$

From Legendre's theorem, we know,
The sum of all the angles of a triangle is equal to two right angles.
$\therefore$ in $△$AOB,
$\angle$ ABO + $\angle$ AOB + $\angle$ BAO = 180${}^{\circ }$
$\Rightarrow$45${}^{\circ }$ + $\angle$ AOB + 75${}^{\circ }$ = 180${}^{\circ }$
$\Rightarrow$ $\angle$ AOB = 60${}^{\circ }$.
But $\angle$ AOB = $\angle$ COD.
(Vertically opposite angles)
$\therefore$ $\angle$ COD = 60 ${}^{\circ }$.
Using Legendre's axiom in $△$COD,
$\angle$ COD + $\angle$ OCD + $\angle$ ODC = 180 ${}^{\circ }$
$\Rightarrow$ 60 ${}^{\circ }$ + 70 ${}^{\circ }$ + $\angle$ ODC = 180 ${}^{\circ }$
$\Rightarrow$ $\angle$ ODC = 50 ${}^{\circ }$