Question

Fill in the blanks :

If two parallel lines are cut by a transversal, then each pair of alternate angles is ……………..

Answer 1

Alternate angles :

As shown in the given figure, two parallel lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ and a transversal $\overleftrightarrow{PS}$ intersecting $\overleftrightarrow{AB}$ at point $Q$ and $\overleftrightarrow{CD}$ at point $R$.

$\angle BQR$, $\angle CRQ$ and $\angle AQR$, $\angle QRD$ are the two pairs of alternate (interior) angles. And $\angle AQP$, $\angle DRS$ and $\angle BQP$, $\angle CRS$ are the two pairs of alternate (exterior) angles.

$\angle AQP=\angle CRQ$ ($\because$Corresponding angles) ……………$\left(i\right)$

And, $\angle AQP=\angle BQR$ ($\because$Vertically opposite angles) ……………$\left(ii\right)$

From eq. $\left(i\right)$ and $\left(ii\right)$, we get :

$\angle BQR=\angle CRQ$

Similarly, $\angle AQR=\angle QRD$, $\angle AQP=\angle DRS$, $\angle BQP=\angle CRS$.

Hence when two parallel lines are cut by a transversal, then each pair of alternate angles is equal.