Question

In the adjoining figure, $BD$ and $CD$ are angle bisectors of $\angle ABC$ and $\angle ACE$ respectively(Line BC is extended to E). Then, which of the following is true?

• A. $\angle D=\frac{1}{2}\angle A$
• B. $\angle x+\angle y=\angle A+\angle D$
• C. $\angle D=\frac{\angle x+\angle y}{2}$
• D. All of above

Answer 1

The correct option is A $\angle D=\frac{1}{2}\angle A$

Extend the line $BC$ to $E$

$BD$ and $CD$ are angular bisectors,

$\therefore \angle ABD=\angle DBC=x$ and $\angle ACD=\angle DCE=y$

$\angle ABC=2xand\angle ACE=2y$

Consider $△ABC$,

$\angle ACE=\angle ABC+\angle BAC$ ------exterior angle is equal to sum of interior opposite angle

$2y=2x+\angle A$

$y-x=\frac{\angle A}{2}$------(i)

Consider $△BCD$,

$\angle DCE=\angle DBC+\angle BDC$ ------exterior angle is equal to sum of interior opposite angle

$y=x+\angle D$

$y-x=\angle D$------(ii)

From(i) and (ii)

$\angle D=\frac{1}{2}\angle A$

Option A. Rest are not true.