Question

In the given figure, AB || CD, ∠BAE = 100° and ∠AEC = 30°. Find ∠DCE.
Figure

Answer 1

$$\begin{gathered} \mathrm{Construction}: \\ D\mathrm{raw}GH\parallel FC\mathbf{\text{as shown in the book}}\mathrm{by}\mathrm{dotted}\mathrm{line}GH, \\ \mathrm{which}\mathrm{meets}AB\mathrm{at}A\mathrm{and}CD\mathrm{at}H. \\ \mathrm{Now}, \\ \angle GAE=\angle AEC=30°\left(\mathrm{alternate}\mathrm{angle}\text{s}\right) \end{gathered}$$
$$\begin{gathered} \mathrm{And}\angle GAB=\angle BAE-\angle GAE \\ =100°-30° \\ =70° \\ \mathrm{Now},AB\parallel CD\mathrm{and}GH\hspace{0.17em}\mathrm{is}\text{the}\mathrm{transversal}. \\ \angle DHA\hspace{0.17em}=\angle GAB=70°\left(\mathrm{corresponding}\mathrm{angles}\right) \\ \mathrm{Again}GH\parallel FC\text{and}DC\mathrm{is}\text{the}\mathrm{transversal}. \\ \therefore \angle DCE=\angle DHA\hspace{0.17em}=70°\left(\mathrm{corresponding}\mathrm{angles}\right) \end{gathered}$$