Every pair of corresponding angles are equal when a line intersects two parallel lines to form their transversal.

The transversal of the two parallel lines forms four pairs of corresponding angles. Each pair of corresponding angles are equal due to same angle of parallel lines and intersection of both parallel lines by the same straight line.

$\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ are two parallel lines and they both are intersected by a straight line $\overleftrightarrow{XY}$. This intersection of them is known as transversal of two parallel lines.

The intersection of straight line $\overleftrightarrow{XY}$ with parallel lines formed four pairs of corresponding angles. Every pair of corresponding angles are equal geometrically.

$P$ and $Q$ are two intersecting points of transversal of the parallel lines.

1

$\angle XPB$ and $\angle XQD$ are one pair of two corresponding angles of transversal of the parallel lines.

The angle of $XPB$ is exactly equal to the angle of $XQD$. The $\angle XPB$ is formed by the intersection of lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{XY}$. The angle $XQD$ is formed by the intersection of lines $\overleftrightarrow{CD}$ and $\overleftrightarrow{XY}$.

The lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ are parallel lines and they both intersected by the same line $\overleftrightarrow{XY}$. Hence, the $\angle XPB$ and $\angle XQD$ are equal in measure.

$\therefore \,\, \angle XPB = \angle XQD$

2

$\angle YPB$ and $\angle YQD$ are another pair of two corresponding angles of transversal of the parallel lines.

The $\angle YPB$ is formed by the intersection of lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{XY}$, and $\angle YQD$ is formed by the intersection of lines $\overleftrightarrow{CD}$ and $\overleftrightarrow{XY}$.

The angles of the lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ are same because they are parallel lines and they both are intersected by the same line $\overleftrightarrow{XY}$. Hence, the angle $YPB$ is exactly equal to the angle $YQD$.

$\therefore \,\, \angle YPB = \angle YQD$

3

$\angle APY$ and $\angle CQY$ are another pair of two corresponding angles, which are formed by the intersection of a line with two parallel lines.

The angle $APY$ is geometrically formed by the intersection of straight lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{XY}$. Similarly, the angle $CQY$ is formed by the intersection of straight lines $\overleftrightarrow{CD}$ and $\overleftrightarrow{XY}$.

Due to the parallelism of straight lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$, and intersection of both lines by same line $\overleftrightarrow{XY}$, the corresponding angles $\angle APY$ and $\angle CQY$ are equal.

$\therefore \,\, \angle APY = \angle CQY$

4

$\angle APX$ and $\angle CQX$ are two corresponding angles, formed by the intersection of a line with two parallel lines.

The angle $APX$ is formed by the intersection of lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{XY}$. In the same way, the angle $CQX$ is formed by the intersection of lines $\overleftrightarrow{CD}$ and $\overleftrightarrow{XY}$.

The $\angle APX$ is exactly equal to $\angle CQX$ because the angles of both straight lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ are same, and they both are intersected by the same line $\overleftrightarrow{XY}$.

$\therefore \,\, \angle APX = \angle CQX$

The equality of each pair of corresponding angles of the transversal of the parallel lines can be proved geometrically.

- Draw two parallel lines of any length with any angle. For example, $\overleftrightarrow{EF}$ and $\overleftrightarrow{GH}$ are two parallel lines and they both are drawn with $10^\circ$ angle.
- Draw an intersecting line of any length with any angle except angle of parallel lines because it becomes another parallel line to them and never intersect it. For example, $\overleftrightarrow{KL}$ is a straight line and it is drawn with $50^\circ$ angle.
- The intersecting line $\overleftrightarrow{KL}$ intersected parallel lines $\overleftrightarrow{EF}$ and $\overleftrightarrow{GH}$ at points $R$ and $S$. Similarly, it formed four pairs of corresponding angles geometrically.

Thus, transversal of two parallel lines is drawn geometrically to study the equality property of the corresponding angles.

$\angle LSF$ and $\angle LRH$ are one pair of two corresponding angles.

The angles of $LSF$ and $LRH$ are unknown but measure them by using a protractor. The angle $LSF$ will be exactly $40^\circ$. The angle $LRH$ will also be $40^\circ$ exactly.

$\angle LSF = \angle LRH = 40^\circ$

$\angle KSF$ and $\angle KRH$ are second pair of two corresponding angles of the transversal of the parallel lines.

The two corresponding angles are unknown and measure them by using a protractor. The $\angle KSF$ will be $140^\circ$ exactly and the $\angle KRH$ will also be $140^\circ$.

$\angle KSF = \angle KRH = 140^\circ$

$\angle KRG$ and $\angle KSE$ are third pair of corresponding angles of the parallel lines transversal.

The angles $KRG$ and $KSE$ are unknown. So, measure them by using a protractor. The $\angle KRG$ will be $40^\circ$ exactly and the $\angle KSE$ will also be $40^\circ$ exactly. The two corresponding angles are equal.

$\angle KRG = \angle KSE = 40^\circ$

$\angle GRL$ and $\angle ESL$ are fourth pair of corresponding angles.

The angles $GRL$ and $ESL$ are unknown but measure them by a protractor. The $\angle GRL$ will be $140^\circ$ exactly and the $\angle ESL$ will also be $140^\circ$ exactly.

$\angle GRL = \angle ESL = 140^\circ$

This example proved that every pair of corresponding angles of the parallel lines transversal are equal.

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