Question : The side BC of a triangle ABC is produced to D. If $\angle ACD = 112^\circ$ and $\angle B =\frac{3}{4} \angle A$ then the measure of $\angle B$ is:
Option 1: $30^\circ$
Option 2: $48^\circ$
Option 3: $45^\circ$
Option 4: $64^\circ$
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Correct Answer: $48^\circ$
Solution :
Assume $\angle BAC=x$
Then, $\angle B=\frac{3x}{4}$
We know that
The sum of two interior angles is equal to the exterior angle of the third angle.
$\angle BAC+\angle B=112^\circ$
⇒ $x+\frac{3x}{4}=112^\circ$
⇒ $\frac{7x}{4}=112^\circ$
⇒ $x=\frac{112×4}{7}=64^\circ$
$\therefore \angle B=\frac{3}{4}×64^\circ=48^\circ$
Hence, the correct answer is $48^\circ$.
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