Question

In the adjoining figure, AB || CD. The lines are cut by a transversal t at E and F, respectively. If ∠2 : ∠1 = 5 : 4, find the measure of each one of the marked angles.

Answer 1

Let $\angle 2=5x\text{and}\angle 1=4x$. Then,
$$\begin{gathered} \angle 1+\angle 2=180°\left[\text{Since AEB is a straight line}\right] \\ \Rightarrow 5x+4x=180° \\ \Rightarrow 9x=180° \\ \Rightarrow x=20° \\ \therefore \angle 1=4\times 20°=80°\text{and}\angle 2=5\times 20°=100° \end{gathered}$$

$$\begin{gathered} \angle 1=\angle 5\left[\text{Corresponding angles}\right] \\ \Rightarrow \angle 5=80° \\ \angle 1=\angle 3\left[\text{Vertically-opposite angles}\right] \\ \Rightarrow \angle 3=80° \end{gathered}$$

$$\begin{gathered} \angle 5=\angle 7\left[\text{Vertically-opposite angles}\right] \\ \Rightarrow \angle 7=80° \end{gathered}$$

$$\begin{gathered} \angle 2=\angle 4\left[\text{Vertically-opposite angles}\right] \\ \Rightarrow \angle 4=100° \\ \angle 2=\angle 6\left[\text{Corresponding angles}\right] \end{gathered}$$

$$\begin{gathered} \Rightarrow \angle 6=100° \\ \angle 6=\angle 8\left[\text{Vertically-opposite angles}\right] \\ \Rightarrow \angle 8=100° \\ \therefore \angle 1=80°\mathbf{,}\mathbf{}\angle 2=100°,\angle 3=80°,\angle 4=100°,\angle 5=80°, \\ \angle 6=100°,\angle 7=80°\text{and}\angle 8=100° \end{gathered}$$