Perpendicular and Parallel Lines

Two lines that intersect to form right angles are called perpendicular. Two lines that never intersect are said to be parallel. Consequently, parallel lines form no angles. However, if a third line, called a transversal, intersects a pair of parallel lines, eight angles are formed, and the relationships between these angles are very important.

Key Fact G5

If a pair of parallel lines is cut by a transversal that is perpendicular to the parallel lines, all eight angles are right angles.

Parallel lines l and m are intersected by perpendicular transversal k, which creates four 90-degree angles where k and l intersect and four 90-degree angles where k and m intersect.

Key Fact G6

If a pair of parallel lines is cut by a transversal that is not perpendicular to the parallel lines:

Four of the angles are acute, and four are obtuse.
All four acute angles are congruent.
All four obtuse angles are congruent.
The sum of the measures of any acute angle and any obtuse angle is 180°.

Parallel lines l and m are intersected by transversal k. At the two intersections, there are the following angles: y and x above the line and x and y below the line. x + y is 180.

Consider the diagram below, in which a pair of parallel lines is crossed by a transversal, creating eight angles.

Parallel lines are intersected by a transversal. The top intersection has four angles: a, b, d, c. The bottom intersection has another four angles: e, f, h, g.

Undoubtedly, in your math class you learned names for various pairs of these angles. Those names won’t appear on the Math 1 test, but referring to them by name may help you remember the various parts of KEY FACT G6.

Angles d and f, as well as angles c and e, are called alternate interior angles because they are interior to the two parallel lines and on alternate sides of the transversal.
Angles a and g, as well as angles b and h, are called alternate exterior angles because they are exterior to the parallel lines and on alternate sides of the transversal.
Pairs a and e, b and f, d and h, and c and g are called corresponding angles because they are in corresponding positions in relationship to the parallel lines and the transversal.
Pairs d and e as well as c and f are called consecutive interior angles because they are interior to the parallel lines and on the same side of the transversal.

When a pair of parallel lines is cut by a transversal, each pair of alternate interior angles, alternate exterior angles, and corresponding angles are congruent. Each pair of consecutive interior angles are supplementary (the sum of their measures is 180°).