Question

In the given figure, prove that:
x = α + β + γ.
Figure

Answer 1

Join BD to produce it to E.
Side BD of triangle ABD is produced to E.
$\therefore \angle ADE=\angle BAD+\angle ABD----\left(1\right)(\mathrm{Exterior}\mathrm{angle}\mathrm{property})$
Side BD of triangle BCD is produced to E.
$\therefore \angle CDE=\angle CBD+\angle BCD----\left(2\right)(\mathrm{Exterior}\mathrm{angle}\mathrm{property})$
$$\begin{gathered} \mathrm{Adding}\left(1\right)\mathrm{and}\left(2\right),\text{we get:} \\ \angle ADE+\angle CDE=\angle BAD+\angle ABD+\angle CBD+\angle BCD \\ \Rightarrow \angle CDA=\angle BAD+\angle BCD+\angle ABD+\angle CBD \\ \Rightarrow x°=\angle BAD+\angle BCD+\angle CBA \\ \Rightarrow x°=\alpha °+\gamma °+\beta ° \\ \therefore x=\alpha +\beta +\gamma . \end{gathered}$$