Question

From the given figure, prove that $$ \angle x+\angle y=\angle A+\angle C $$

Answer 1

Construction: Join A to $$ \mathrm{C} $$
$$ \ln \Delta \mathrm{ABC} $$
$$ \angle \mathrm{x}=\angle 1+\angle 2 \ldots . $$ (i) (exterior angle is equal to sum of opposite interior angles)
$$ \mathrm{Also} $$ in $$ \Delta \mathrm{ADC} $$
$$ \angle y=\angle 3+\angle 4 \ldots $$ (ii) (reason as above)
Adding (i) and (ii), we get
$$ \angle x+\angle y=\angle 1+\angle 2+\angle 3+\angle 4 $$
$$ \Rightarrow \angle x+\angle y=(\angle 1+\angle 3)+(\angle 2+\angle 4) $$
Hence, $$ \angle x+\angle y=\angle A+\angle C $$
Hence proved