Question

In the adjoining figure, AB || CD. The lines are cut by a transversal t at E and F, respectively. If ∠1 = 70°, find the measure of each of the remaining marked angles.

Answer 1

We have, $\angle 1=70°$. Then,
$$\begin{gathered} \angle 1=\angle 5\left[\text{Corresponding angles}\right] \\ \Rightarrow \angle 5=70° \\ \angle 1=\angle 3\left[\text{Vertically-opposite angles}\right] \\ \Rightarrow \angle 3=70° \end{gathered}$$

$$\begin{gathered} \angle 5=\angle 7\left[\text{Vertically-opposite angles}\right] \\ \Rightarrow \angle 7=70° \\ \angle 1+\angle 2=180°\left[\text{Since AFB is a straight line}\right] \\ \Rightarrow 70°+\angle 2=180° \end{gathered}$$

$$\begin{gathered} \Rightarrow \angle 2=110° \\ \angle 2=\angle 4\left[\text{Vertically-opposite angles}\right] \\ \Rightarrow \angle 4=110° \\ \angle 2=\angle 6\left[\text{Corresponding angles}\right] \end{gathered}$$

$$\begin{gathered} \Rightarrow \angle 6=110° \\ \angle 6=\angle 8\left[\text{Vertically-opposite angles}\right] \\ \Rightarrow \angle 8=110° \\ \therefore \angle 1=70°\mathbf{,}\mathbf{}\angle 2=110°,\angle 3=70°,\angle 4=110°,\angle 5=70°, \\ \angle 6=110°,\angle 7=70°\text{and}\angle 8=110° \end{gathered}$$