Question

The exterior angles obtained on producing the base of a triangle both ways are ${104}^{\circ }$ and ${136}^{\circ }$ .Find all the angles of the triangle.

Answer 1

In $\Delta ABC$, base BC is produced both ways to D and E respectively forming $\angle ABE={104}^{\circ }$ and $\angle ACD={136}^{\circ }$.

$\therefore \angle ABC+\angle ABE={180}^{\circ }$ (Linear pair)

$\Rightarrow \angle ABC+{104}^{\circ }={180}^{\circ }$
$\Rightarrow \angle ABC={180}^{\circ }-{104}^{\circ }$
$\therefore \angle ABC={76}^{\circ }$
Similarly, $\angle ACB+\angle ACD={180}^{\circ }$
$\Rightarrow \angle ACB+{136}^{\circ }={180}^{\circ }$
$\Rightarrow \angle ACB={180}^{\circ }-{136}^{\circ }={44}^{\circ }$
$\text{B}ut\angle ABC+\angle ACB+\angle BAC={180}^{\circ }$
(Angles of a triangle)
$\Rightarrow {76}^{\circ }+{44}^{\circ }+\angle BAC={180}^{\circ }$
$\Rightarrow {120}^{\circ }+\angle BAC={180}^{\circ }$
$\Rightarrow \angle BAC={180}^{\circ }-{120}^{\circ }={60}^{\circ }$
$\text{H}enceanglesofthetriangleare$
${60}^{\circ },{76}^{\circ },{44}^{\circ }$