Question

In $∆\mathrm{ABC},\angle \mathrm{{\rm A}}=100°,\mathrm{AD}$ bisects $\angle \mathrm{A}$ and $\mathrm{AD}\perp \mathrm{BC}$. Then, $\angle \mathrm{B}$ is equal to

• A. $80°$
• B. $20°$
• C. $40°$
• D. $30°$

Answer 1

The correct option is C

$40°$

Step 1: Calculate $\angle \mathrm{BAD}$

Let us construct the triangle.

Given: $\angle \mathrm{A}=100°$

$\mathrm{AD}$ bisects $\angle \mathrm{A}$.

$\therefore \angle \mathrm{BAD}=\frac{100}{2}$

$=50°$

Step 2: Calculate $\angle B$

We know that the sum of the angles of a triangle is $180°$.

Therefore, for $△\mathrm{ABD}$,

$\angle \mathrm{ABD}+\angle \mathrm{BAD}+\angle \mathrm{ADB}=180°$

$\Rightarrow$ $\angle \mathrm{ABD}+50+90=180$ [Given that $\mathrm{AD}\perp \mathrm{BC}$, $\therefore \angle \mathrm{ADB}=90°$]

$\Rightarrow$ $\angle \mathrm{ABD}+140=180$

$\Rightarrow$ $\angle \mathrm{ABD}=180-140$

$\Rightarrow$ $\angle \mathrm{ABD}=40°$

Hence, $\angle \mathrm{B}$ is equal to $40°$ and so option (C) is the correct answer.