# How do you prove corresponding angles?

The proof is based on the *Fifth Postulate of Euclid*.

See below.

Let

Two *one-sided* angles (on one side of transverse *interior* or *inner* (between the parallels) and another *exterior* (outside of the area between the parallel lines), are called *corresponding*.

So, angles

Take into account that sum of measures of *supplemental* angles

That is,

Now, if the *corresponding* angles *inner* angles

There are two cases, both mean that a sum of some pair of *inner* angles measure less than

*inner* angles' measures is less than

Let's recall the *Fifth Postulate of Euclid*.

If two lines on a plane intersect a third line in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

Therefore, since

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Corresponding angles are proved to be equal when two parallel lines are cut by a transversal. This can be shown using the property that when two parallel lines are cut by a transversal, the alternate interior angles, alternate exterior angles, corresponding angles, and consecutive interior angles are congruent.

To prove corresponding angles are equal:

- Begin with two parallel lines cut by a transversal, forming corresponding angles.
- Identify corresponding angles by locating angles on the same side of the transversal and on the same side of the parallel lines.
- Use the property of corresponding angles, which states that corresponding angles are equal.
- State the reasoning for the equality of the corresponding angles, citing the property of parallel lines cut by a transversal.
- Conclude by stating that the corresponding angles are equal due to the given properties of parallel lines and transversals.

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