Plane Geometry, Parallel Lines and Interior Angles

Parallel Lines and Interior/Exterior Angles

Two lines are parallel if you can draw a third line perpendicular to both. This is a restatement of Euclid's fifth postulate, which we will accept for now. It's true in the plane, which is where geometry usually takes place.

Let l1 and l2 be parallel horizontal lines, and let lv be a vertical line, perpendicular to l1 and l2. Draw lu to the left of lv perpendicular to l1. Since l1 and l2 never meet, the angles on either side of lu, between l1 and l2, sum to 180°. Thus all angles are right angles, and lu is perpendicular to l2. We have drawn a rectangle.

If the picture were reflected through a mirror, we would still have four right angles, and opposite pairs of lines would still be parallel. If the left side of the rectangle were shorter than the right, then by symmetry the right side should be shorter than the left. The same assumptions hold in either direction. This is a contradiction, hence the left and right sides of the rectangle have the same length, and the top and bottom have the same length.

parallel lines with a slanted line cutting across

Let ls be a slanted line from lower left to upper right. This forms a diagonal through the rectangle. The triangle above and the triangle below have sides that are exactly the same length, hence the triangles, and their corresponding angles, are congruent by SSS.

The acute angle above l1 and to the right of ls is called an interior angle, and the obtuse angle above l1 and to the left of ls, which contains the left side of the rectangle, is called an exterior angle. These angles are supplementary, as they form l1. They are also congruent to the interior and exterior angles below l2, to the left and right of ls respectively. Thus ls cuts across l1 and l2, and creates congruent alternate interior angles and congruent alternate exterior angles.

This leads to another equivalent definition of parallel. Two lines l1 and l2 are parallel if a (possibly slanted) line ls crosses l1 and l2, and creates congruent alternate interior angles, or congruent alternate exterior angles. Let x and y be the angles to the right of ls, above l1 and below l2. Let z be the angle to the left of ls, above l1. Now x and z are supplementary, and z = y, hence x and y are supplementary. Our lines do not join at the right. Similarly, the lines do not join at the left; they are parallel.

The angles to the right of ls, and above l1 and l2 are called corresponding angles. These are equal iff l1 and l2 are parallel.