Question

In the above figure, $D E F$ is a triangle whose side $D F$ is produced to $G . H F$ and $H D$ are the bisectors of $\angle E F G$ and $\angle E D G$, respectively. If $\angle D E F=23 \frac{1}{2}^{\circ}$, then $\angle D H F$ (in degrees) $=$

• A. $11 \frac{1}{2}$
• B. $11 \frac{2}{5}$
• C. $11 \frac{3}{4}$
• D. $11 \frac{1}{3}$

Answer 1

Answer: (C)

(i) $DH$ and $FH$ are angle bisectors. Exterior angle of a triangle equals the sum of its interior opposite angles.
(ii) Take $\angle F D H=\angle H D E=x$ and $\angle E F H=\angle H F G$ $=\gamma$.
(iii) $\angle E F G$ is the exterior angle of $\triangle D E F$.
(iv) $\angle D E F=23 \frac{1}{2}^{\circ}$ and $2 x+y=180^{\circ}-23 \frac{1}{2}^{\circ}$
(v) Find $x$ and $y$ and $H$.
$\therefore 2 y=2 x+23 \frac{1}{2}$
Simplify to obtain the value of the required angle.