Question

If the side BC of $\Delta ABC$ is produced on both sides, then write the difference between the sum of the exterior angles so formed and $\angle A.$

Answer 1

In $\Delta ABC,$ side BC is produced on both sides forming exterior $\angle ABE$ and $\angle ACD$

Ext. $\angle ABE=\angle A+\angle ACB$

and Ext. $\angle ACD=\angle ABC+\angle A$

Adding we get,

$\angle ABE+\angle ACD=\angle A+\angle ACB+\angle A+\angle ABC$

$\Rightarrow \angle ABE+\angle ACD-\angle A=\angle A+\angle ACB+\angle A+\angle ABC-\angle A$ (Subtracting $\angle A$ from both sides)

$\Rightarrow$ $\angle$ ABE+$\angle$ ACD - $\angle$ A =$\angle A+\angle ABC+\angle ACB$

$\Rightarrow$ $\angle$ ABE+$\angle$ ACD - $\angle$ A =$\angle A+\angle B+\angle C={180}^{\circ }$ (Sum of angles of a triangle)

Hence, difference between the sum of the exterior angles so formed and $\angle A$ is ${180}^{\circ }$.