Question

The sides BA and DC of a quadrilateral are extended as shown in figure 5.11. Prove that m + n = p+q

Answer 1

$\angle$EAD and $\angle$DAB make a linear pair of angles.
Thus, we have:
$\angle$EAD + $\angle$DAB = 180°
$\Rightarrow$ n + $\angle$DAB = 180°
$\Rightarrow$ $\angle$DAB = 180° – n …(1)

Also, $\angle$BCF and $\angle$BCD make a linear pair of angles.
Thus, we have:
$\angle$BCF + $\angle$BCD = 180°
$\Rightarrow$ m + $\angle$BCD = 180°
$\Rightarrow$ $\angle$BCD = 180° – m …(2)

Now, by angle addition property of a quadrilateral,
$\angle$DAB + $\angle$ABC + $\angle$BCD + $\angle$CDA = 360°
$\Rightarrow$180° – n + q + 180° – m + p = 360°
$\Rightarrow$ p + q – m – n + 360° = 360°
$\Rightarrow$ p + q = m + n
$\Rightarrow$m + n = p + q