Corresponding Angles

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Consider two parallel lines cut by a transversal as shown below: 

Two parallel lines cut by transversal

We have four pairs of corresponding angles. What is the relation between each pair? The angles are equal in each pair. This is a very basic truth of the Euclidean system, and therefore it is taken as an axiom.

Corresponding Angles Axiom: If a transversal cuts two parallel lines, then each pair of corresponding angles is equal.

Now, suppose that a transversal cuts two lines such that a pair of corresponding angles is equal, as shown below:

Corresponding angles axiom

What can we say about the two lines? Can we say that they are parallel? If the two lines were not parallel, these corresponding angles could not have been equal. Thus, these lines must be parallel.

Axiom: If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel to each other.

We can think of this converse as a part of the corresponding angles axiom itself. And finally, what if two corresponding angles are not equal? Can the lines be parallel. The answer is no! Corresponding angles can be equal only if the lines on which they are formed are parallel.

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