Important Formulas: Lines and Angles

Basic definitions and facts

• Transversal. A line which intersects two or more given lines at distinct points is called a transversal to those lines.
• Parallel lines. Two lines in a plane are parallel if they do not meet (that is, they do not intersect) when produced indefinitely in either direction.
• Distance between lines. The distance between two intersecting lines is zero. The distance between two parallel lines is the same at every point; it is equal to the length of a common perpendicular (the perpendicular distance) between them.

Angles formed by a transversal with two parallel lines

When two parallel lines are cut by a transversal, the following angle relationships always hold:

• Alternate interior (or exterior) angles are equal.
• Corresponding angles are equal.
• Interior angles on the same side of the transversal are supplementary (their sum is 180°).

When the lines are not parallel

If two lines are not parallel (that is, they meet at some point) and are cut by a transversal, then the equalities or supplementary relations listed above need not hold in general.

Converse tests for parallelism

If two lines are cut by a transversal, then the lines are parallel if any one of the following conditions is true:

• The corresponding angles are equal.
• The alternate interior angles are equal.
• The interior angles on the same side of the transversal are supplementary.

Useful remarks and terminology

• Corresponding angles: One angle is formed by one line and the transversal and the other is the angle in the corresponding position formed by the other line and the transversal.
• Alternate interior angles: Angles that lie between the two lines and on opposite sides of the transversal.
• Interior angles on the same side of the transversal: Angles that lie between the two lines and on the same side of the transversal; these add to 180° when the lines are parallel.
• Perpendicular distance: The shortest distance between two parallel lines; it is the length of any segment perpendicular to both lines.

Applications and quick checks

• To test whether two lines are parallel, check any one of the converse conditions: equal corresponding angles, equal alternate interior angles, or supplementary interior angles on the same side of the transversal.
• When solving geometry problems, mark equal angles and supplementary pairs clearly; use them to set up equations when measures are unknown.
• Remember that if a figure states two lines are parallel, you may immediately set up equalities and supplementary relations for angle chasing; conversely, if you can prove one of the relations, you may conclude parallelism.

Summary

Transversals create several standard angle relationships with two lines. These relationships-equality of alternate interior and corresponding angles, and supplementarity of interior angles on the same side are both consequences of parallelism and tests for it. The perpendicular distance between parallel lines is constant, and intersecting lines have distance zero.