When a transversal (any line that passes through two parallel lines) cuts two parallel lines then the angles that correspond to the similar positions (angles in the matching corners) with respect to either of the parallel lines and lie on the same side of the transversal are called the Corresponding angles. The angles lie on the same side of the parallel lines.
Corresponding Angles
In this figure, Line m and line n are parallel. The transversal cuts both the lines. $\angle$b and $\angle$f are angles formed when the transversal cuts m and n and these are one pair of angles that correspond to the similar position (above m and above n) and lie on the same side of the transversal. Thus $\angle$b and $\angle$f are corresponding angles. There are many other pairs of corresponding angles in this figure. $\angle$a and $\angle$e; $\angle$c and $\angle$g; $\angle$d and $\angle$h are the other pairs of corresponding angles in this figure.

To identify the corresponding angles it is easy by writing the letter F along the two parallel lines and to identify the angles around the corners of the two horizontal lines of F and the transversal. The letter F can be written on either ways on the parallel lines.
Corresponding Angle

If two parallel lines are cut by a transversal, then the pairs of corresponding angles are always congruent.Postulates are not proved, but are assumed to be self-evident and true. So pairs of angles which lie on the same side of transversal and the same side of both parallel lines are congruent.

Corresponding Angles Postulates

In this figure, two lines m and n are cut by the transversal l,
if m || n, then $\angle$1 $\cong$ $\angle$2
If any two lines are cut be a transversal and the corresponding angles are congruent, then the two lines are parallel.
Angles which are at the same location at each intersection when a transversal cuts any two lines are equal, then the two lines are parallel.
Converse of Corresponding Angle Postulate
This means if the $\angle$1 = $\angle$2, which are at the same location at the intersection when a transversal l cuts the two lines m and n, then the two lines m and n are parallel.
In a congruent triangle, every part of it is also congruent. If two triangles are congruent, then all the sides and angles are also congruent with their corresponding pair. Such pairs of angles are called Congruent Corresponding Angles.
In every congruent triangle, there are 3 sets of congruent angles.Consider the two congruent triangles $\Delta$DEF and $\Delta$GHI shown below. There are three pairs of corresponding congruent angles. The corresponding congruent angles are marked with similar arcs and similar colour.
Congruent Corresponding Angles
$\angle$D $\cong$ $\angle$G.
$\angle$E $\cong$ $\angle$H.
$\angle$F $\cong$ $\angle$I.
Two lines cut by a transversal are parallel if and only if corresponding angles are congruent.

Proof: Suppose that L, M and N are three distinct lines.

Corresponding Angles Theorem

Then we need to prove that M and N are parallel if and only if corresponding angles of the intersection of M and L, and N and L are equal.

First we assume L and M are parallel and prove that corresponding angles are equal.

Assuming M || N, label a pair of corresponding angles as $\angle$a and $\angle$b. We know that $\angle$c is supplementary to $\angle$a as they form a linear pair (because L is a line, and any point on L can be considered as a straight angle between two points on either side of the point). Note that $\angle$b and $\angle$c are also supplementary, since they form interior angles of parallel lines on the same side of the transversal L.
Therefore, $\angle$c = 180° - $\angle$a = 180° – $\angle$bThus we have, $\angle$a = $\angle$b.

This can be proven for every pair of corresponding angles in the same way as, mentioned above.

Now, we need to prove the converse by assuming that corresponding angles are equal and prove M and N are parallel.

Corresponding Angle Theorem
Assuming corresponding angles, label each $\angle$a and $\angle$b appropriately. By the straight angle theorem, we can label every corresponding angle either $\angle$a or $\angle$b.
For example, we know $\angle$a + $\angle$b = 180° on the right side of the intersection of M and L, since it forms a straight angle on L. Consequently, we can label the angles on the left side of the intersection of M and L as $\angle$a or $\angle$b since they form straight angles on M. Thus, we have, $\angle$a + $\angle$b = 180°, we know that the interior angles on either side of L add up to 180°. By the same side interior angles theorem, this makes M || N.

Hence proving that “Two lines cut by a transversal are parallel if and only if corresponding angles are congruent. “
A window pane is a good example to understand different pairs of angles.

In this picture of window pane $\angle$a and $\angle$b are corresponding angles.
Corresponding Angles in Real life

In this key the angles marked in red are corresponding angles.
Corresponding Angle in Real Life

Solved Example

Question: In the given figure below, if the $\angle$a is 72$^o$, then find all the other angles.
 
Corresponding Angles Example
Solution:
 
Given that $\angle$a = 72$^o$.

By definition of corresponding angles we have, $\angle$e and $\angle$a are corresponding angles.

Also, By Corresponding Angles Postulate,  $\angle$a and $\angle$e are congruent and therefore equal. Thus $\angle$e = 72$^o$.

In Geometry, We know that a straight angle has a measure of 180$^o$.  So, $\angle$a + $\angle$d = 180$^o$.
72 + $\angle$d = 180$^o$
$\angle$d = 180 - 72
$\angle$d = 108$^o$

Here $\angle$d and $\angle$h are also corresponding angles.  By Corresponding Angles Postulate , $\angle$d and $\angle$h are congruent and therefore equal. Thus, $\angle$h = 108$^o$.

Again, we know that, a straight angle has a measure of 180$^o$. So, $\angle$a + $\angle$b = 180$^o$.
72 + $\angle$b = 180$^o$
$\angle$b = 180 - 72
$\angle$b = 108$^o$

Here, $\angle$b and $\angle$f are corresponding angles. By Corresponding Angles Postulate, $\angle$b and $\angle$f are congruent and therefore equal. Thus, $\angle$f = 108$^o$.

We know that a straight angle has a measure of 180 degree. So, $\angle$b + $\angle$c = 180$^o$.
108 + $\angle$c = 180$^o$
$\angle$c = 180 – 108
$\angle$c = 72$^o$

Here also, $\angle$c and $\angle$g are corresponding angles.

Thus, $\angle$c and $\angle$g are congruent and therefore equal. Thus, $\angle$g = 72$^o$.

Therefore, $\angle$a = $\angle$e = $\angle$c = $\angle$g = 72$^o$.

$\angle$b = $\angle$f = $\angle$d = $\angle$h = 108$^o$.