Geometry Chapter 3: Parallel and Perpendicular Lines

Skew Lines

Lines that do not intersect and are not coplanar.

Parallel Lines

lines in the same plane that never intersect

Parallel Planes

Planes that do not intersect

Postulate 3.1: Parallel Postulate

If there is a line and a point not on the line then there is exactly one line through the point parallel to the given line.

Postulate 3.2: Perpendicular Postulate

If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line.

Transversal

A line that intersects two or more coplanar lines at different points

Corresponding Angles

Angles formed by a transversal cutting through 2 or more lines that are in the same relative position.

Alternate Interior Angles

Angles that lie within a pair of lines and on opposite side of a transversal.

Alternate Exterior Angles

Angles that lie outside a pair of lines and on opposite sides of a transversal.

Consecutive Interior Angles

Two angles that lie between the two lines on the same side of the transversal.

Theorem 3.1: Corresponding Angles Theorem

If two parallel lines are cut by a transversal, then the pair of corresponding angles are congruent.

Theorem 3.2: Alternate Interior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

Theorem 3.3: Alternate Exterior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.

Theorem 3.4: Consecutive Interior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.

Theorem 3.5: Corresponding Angles Converse

If two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel.

Theorem 3.6: Alternate Interior Angles Converse

If two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel.

Theorem 3.7: Alternate Exterior Angles Converse

If two lines are cut by a transversal so the alternate exterior angles are congruent, then the lines are parallel.

Theorem 3.8: Consecutive Interior Angles Converse

If two lines are cut by a transversal so the consecutive interior angles are supplementary, then the lines are parallel.

Theorem 3.9: Transitive Property of Parallel Lines

If two lines are parallel to the same line, then they are parallel to each other.

Distance from a Point to a Line

The length of the perpendicular segment from the point to the line.

Perpendicular Bisector

A segment, ray, line, or plane that is perpendicular to a segment at its midpoint

Theorem 3.10: Linear Pair Perpendicular Theorem

If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.

Theorem 3.11: Perpendicular Transversal Theorem

In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line.

Theorem 3.12: Lines Perpendicular to a Transversal Theorem

In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.

Directed Line Segment

A segment, AB, that represents moving from point A to point B.

Theorem 3.13: Slopes of Parallel Lines

In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope.

Theorem 3.14: Slopes of Perpendicular Lines

In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1.