Parallel Lines, Transversals, Angle Pair Relationships

Overview

When two lines are intersected by a transversal, eight angles are formed — four at each intersection. Those angles come in pairs, and each pair has a name based on its position: whether the angles lie between the two lines or outside them, whether they are on the same side of the transversal or on opposite sides, and whether they occupy the same relative position at their intersection. When the two lines are parallel, the named pairs have special properties: some pairs are congruent (equal in measure) and others are supplementary (their measures add to 180°). Those properties let you find unknown angle measures, and solve for x when the measures are given as expressions.

Vocabulary

Parallel Lines: Lines that lie in the same plane and do not intersect. Parallel lines are marked in a figure with matching arrowheads, and they always run in the same direction.
Skew Lines: Lines that do not intersect and are not in the same plane. Skew lines occur in three-dimensional figures and always run in different directions.
Transversal: A line that intersects two or more other lines at different points, all in the same plane.
Interior Angles / Exterior Angles: Interior angles lie between the two lines intersected by the transversal; exterior angles lie outside them.
Corresponding Angles: Two angles in the same relative position at each intersection — for example, the upper-left angle at the top intersection and the upper-left angle at the bottom intersection.
Alternate Interior Angles: Two angles that are interior and on opposite sides of the transversal.
Alternate Exterior Angles: Two angles that are exterior and on opposite sides of the transversal.
Consecutive Interior Angles: Two angles that are interior and on the same side of the transversal. Also called same-side interior angles.
Vertical Angles: Two angles opposite each other at a single intersection. Vertical angles are always congruent, whether or not any lines are parallel.
Linear Pair: Two adjacent angles at a single intersection whose measures add to 180°. A linear pair is always supplementary, whether or not any lines are parallel.

Main Ideas

An angle pair is named by position: interior or exterior, same side or opposite sides of the transversal, and same relative position at each intersection. Angles are numbered 1–8 in the notes and in the interactive panel for this lesson.
If two parallel lines are intersected by a transversal, then corresponding angles, alternate interior angles, and alternate exterior angles are congruent (equal in measure).
If two parallel lines are intersected by a transversal, then consecutive interior angles are supplementary (their measures add to 180°).
Vertical angles and linear pairs live at a single intersection, so they hold whether or not the lines are parallel.
Through any point not on a line, exactly one line can be drawn parallel to the given line, and exactly one line can be drawn perpendicular to it.
To solve for x, first name the pair: set the expressions equal if the pair is congruent, or set their sum to 180° if the pair is supplementary — then check by substituting back.