# Equality of Interior alternate angles of Transversal of Parallel lines

## Property

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.

### Proof

$\overleftrightarrow{AB}$ 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

#### One pair of Interior alternate angles

$\angle APO$ 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

#### Other pair of Interior alternate angles

$\angle OPB$ 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 $\overrightarrow{OY}$.

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

##### Example

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

1. 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.
2. 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.

###### Equality of One pair of Interior alternate 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.

###### Equality of Second pair of Interior alternate angles

$\angle ILF$ 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.