Question

If two interior angles on the same sides of a transversal intersecting two parallel lines are in the ratio $2:3$ , then the smaller of the two angles is

Answer 1

Solve for the value of a smaller angle:

Let $l,m$ be two parallel lines and $n$ be the transversal intersecting the parallel lines

$\angle A,\angle C$ are the angles on the left side of the transversal and

$\angle B,\angle D$are the angles on the right side of the transversal

The alternate angles inside the parallel lines will be equal.

i,e. $\angle A=\angle D$and $\angle B=\angle C$

Let the constant of proportionality be $x$

Hence, the interior angles on the same side of the transversal are $2x$ and $3x$

We know that sum of interior angles on the same side of transversal intersecting two parallel lines is $180º$

Consider right side of the transversal

$\angle B+\angle D={180}^{\circ }$

$$\begin{gathered} \Rightarrow 2x+3x=180º \\ \Rightarrow 5x=180º \\ \Rightarrow x=36º \end{gathered}$$

So the angles are

$$\begin{gathered} 2x=2\times 36º=72º \\ 3x=3\times 36º=108º \end{gathered}$$

Hence, the smaller angle is $72º$