Interior Angles
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.