ABCD is a quadrilateral in which AB || DC and AD = BC. Prove that… - Vidyadip

Question

ABCD is a quadrilateral in which AB || DC and AD = BC. Prove that $\angle\text{A}=\angle\text{B}$ and $\angle\text{C}=\angle\text{D}.$

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Answer

ABCD is a quadrilaateral such that AB || DC and AD = BC

Construction Extend AB to E and draw a line CE parallet to AD. proos since, AD || CE and transverrsal AE cuts then at A and E repectively. $\therefore\angle\text{A}+\angle\text{E}=180^\circ$ [since, sum of cointerios angles is 180°] $\Rightarrow\ \angle\text{A}=180^\circ-\angle\text{E}\ ...(\text{i})$ So, quadrilateral AECD is a parallelogram. $\Rightarrow\ \text{AD}=\text{CE}\Rightarrow\text{BC}=\text{CE}$ [$\because\ \text{AD}=\text{BC}$ given] Now, in $\Delta\text{BCE}$ $\text{CE}=\text{BC}$ [proved above] $\Rightarrow\ \angle\text{CBE}=\angle\text{CEB}$ [opposite angles of equal side are equal] $\Rightarrow\ 180^\circ-\angle\text{B}=\angle\text{E}$ $[\because\angle\text{B}+\angle\text{CBE}=180^\circ]$ $\Rightarrow\ 180^\circ-\angle\text{E}=\angle\text{B}\ ...(\text{ii})$ From Eqs. (i) and (ii) $\angle\text{A}=\angle\text{B}$ Hence proved.

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