Question

In Fig. AD = BC and AD || BC. Is AB = DC? Give reasons in support of your answer.

Figure

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ AD=BC \\ AD\parallel BC \\ \mathrm{We}\mathrm{have}\mathrm{to}\mathrm{show}\mathrm{that}AB=DC. \\ \mathrm{Proof}: \\ AD\parallel BC \\ \therefore \angle BCA=\angle DAC(\mathrm{alternate}\mathrm{angles}) \\ \mathrm{In}△ABC\mathrm{and}△CDA: \\ BC=DA(\mathrm{given}) \\ \angle \mathrm{BCA}=\angle \mathrm{DAC}(\mathrm{proved}\mathrm{above}) \\ \mathrm{AC}=\mathrm{AC}\left(\mathrm{common}\right) \\ By\mathrm{SAS}c\mathrm{ongruence}\mathrm{property}: \\ △ABC\cong △CDA \\ =>AB=CD(\mathrm{corresponding}\mathrm{parts}\mathrm{of}\mathrm{the}\mathrm{congruent}\mathrm{triangles}) \end{gathered}$$