Two pairs of interior alternate angles are formed when a transversal intersects the parallel lines but the pair of interior alternate angles are equal geometrically.

$\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ are two parallel lines and their transversal line is $\overleftrightarrow{XY}$. $P$ and $Q$ are intersecting points of $\overleftrightarrow{XY}$ and $O$ is the exact middle point of the intersecting line $\overleftrightarrow{XY}$.

Total four interior angles are formed due to the intersection of $\overleftrightarrow{XY}$ with parallel lines. $\angle APO$, $\angle OPB$, $\angle OQC$ and $\angle OQD$ are four interior angles.

The four interior angles are classified into two pairs of interior alternate angles.

1

$\angle APO$ and $\angle OQD$ are one pair of interior alternate angles. The two interior alternate angles are equal in measure.

$\angle APO$ is the interior angle formed by the intersection of the lines $\overleftrightarrow{AB}$ and $\overrightarrow{OX}$. The inverted reflexion of the intersection of lines $\overleftrightarrow{AB}$ and $\overrightarrow{OX}$ transforms the interior angle $\angle APO$ into $\angle OQD$ because $\overleftrightarrow{CD}$ is the inverted reflexion of $\overleftrightarrow{AB}$ and $\overrightarrow{OX}$ is the inverted reflexion of $\overrightarrow{OY}$.

$\therefore \,\, \angle APO = \angle OQD$

2

$\angle OPB$ and $\angle OQC$ are another pair of interior alternate angles and these interior alternate angles are also equal in measure.

$\angle OPB$ is another interior angle formed by the intersection of lines $\overleftrightarrow{AB}$ and $\overrightarrow{OX}$. The inverted reflexion of the intersection of lines $\overleftrightarrow{AB}$ and $\overrightarrow{OX}$ transforms the interior angle $\angle OPB$ into $\angle OQC$ because $\overleftrightarrow{CD}$ is the inverted reflexion of $\overleftrightarrow{AB}$ and $\overrightarrow{OX}$ is the inverted reflexion of $\overrightarrow{OY}$.

$\therefore \,\, \angle OPB = \angle OQC$

The equality property of the interior alternate angles of transversal of parallel lines can be proved in geometric system.

- Take ruler and draw two parallel lines of any length with any angle. For example, $\overleftrightarrow{EF}$ and $\overleftrightarrow{GH}$ are two parallel lines and drawn with $20^\circ$ angle.
- Draw an intersecting line with any angle and it intersects both parallel lines at two points. Here, $\overleftrightarrow{IJ}$ is the straight line which is drawn with $60^\circ$ angle and it intersected both parallel lines at points $L$ and $M$.

The geometrical process formed four interior angles and they can be classified as two pairs of interior alternate angles. In this example, $\angle ELI$, $\angle ILF$, $\angle GMJ$ and $\angle JMH$ are four interior angles.

$\angle ELI$ and $\angle JMH$ are two interior alternate angles. Measure each angle by using protractor. The interior angle ($\angle ELI$) will be $40^\circ$ exactly and the interior angle of $\angle JMH$ will also be $40^\circ$. Therefore, the interior alternate angles are equal.

$\angle ILF$ and $\angle GMJ$ are another two interior alternate angles. Measure each angle by using protractor. The interior angle ($\angle ILF$) will be $140^\circ$ exactly and the interior angle ($\angle GMJ$) will also be $140^\circ$ exactly. It means, the interior alternate angles are also equal.

Geometrically, it is proved that interior alternate angles are always equal in the case of transversal of two parallel lines.

Copyright © 2012 - 2017 Math Doubts, All Rights Reserved