# Interior Angles

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Suppose that two parallel lines are intersected by a transversal, as shown below:

What is the relation between any pair of alternate interior angles?

Theorem: When a transversal intersects two parallel lines, each pair of alternate interior angles is equal. Conversely, if a transversal intersects two lines such that a pair of interior angles is equal, then the two lines are parallel.

Proof: Refer to the figure above. We have:

$$\angle 1$$ = $$\angle 5$$ (corresponding angles)

$$\angle 3$$ = $$\angle 5$$ (vertically opposite angles)

Thus,

$$\angle 1$$ = $$\angle 3$$

Similarly, we can prove that $$\angle 2$$ = $$\angle4$$. Conversely, suppose that $$\angle 1$$ = $$\angle 3$$. We have to prove that the lines are parallel. Since $$\angle 3$$ = $$\angle 5$$ (vertically opposite angles), we have:

$$\angle 1$$ = $$\angle 5$$

Thus, a pair of corresponding angles is equal, which can only happen if the two lines are parallel.

What about any pair of co-interior angles?

Theorem: If a transversal intersects two parallel lines, then each pair of co-interior angles is supplementary (their sum is 1800). Conversely, if a transversal intersects two lines such that a pair of co-interior angles is supplementary, then the two lines are parallel.

Proof: Refer to the following figure once again:

We have:

$$\angle 1$$ = $$\angle 5$$ (corresponding angles)

$$\angle 5$$ + $$\angle4$$ = 1800 (linear pair)

è $$\angle 1$$ + $$\angle4$$ = 1800

Similarly, we can show that

$$\angle 2$$ + $$\angle 3$$ = 1800

The converse part of the proof is left to you as an exercise.