Geometry Chapter 3: Lines and Planes

Skew and Parallel Lines

• Skew Lines: Lines that do not intersect and are not in the same plane.
• Parallel Lines: Lines in the same plane that never meet or intersect.
• Parallel Planes: Planes that do not intersect each other.

Postulates

• Parallel Postulate: Given a line and a point not on it, there is exactly one line through the point that is parallel to the original line.
• Perpendicular Postulate: Given a line and a point not on it, there is exactly one line through the point that is perpendicular to the original line.

Transversal and Angles

• Transversal: A line that intersects two or more coplanar lines at different points.
• Corresponding Angles: Angles formed by a transversal that are located in the same relative position with respect to the intersected lines.
• Alternate Interior Angles: Angles that lie between two lines and are on opposite sides of a transversal.
• Alternate Exterior Angles: Angles located outside two lines and on opposite sides of a transversal.
• Consecutive Interior Angles: Two angles located between two intersected lines and on the same side of a transversal.

Angle Theorems

• Corresponding Angles Theorem: If two parallel lines are cut by a transversal, corresponding angles are congruent.
• Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, alternate interior angles are congruent.
• Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, alternate exterior angles are congruent.
• Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, pairs of consecutive interior angles are supplementary.

Converse Theorems

• Corresponding Angles Converse: If two lines are cut by a transversal and the corresponding angles are congruent, the lines are parallel.
• Alternate Interior Angles Converse: If two lines are cut by a transversal and alternate interior angles are congruent, the lines are parallel.
• Alternate Exterior Angles Converse: If two lines are cut by a transversal and alternate exterior angles are congruent, the lines are parallel.
• Consecutive Interior Angles Converse: If two lines are cut by a transversal and consecutive interior angles are supplementary, the lines are parallel.

Additional Theorems

• Transitive Property of Parallel Lines: If two lines are both parallel to the same line, then they are parallel to each other.
• Distance from a Point to a Line: Defined as the length of the perpendicular segment from a point to a line.
• Perpendicular Bisector: A line or segment that divides another segment into two equal parts at a right angle.

Perpendicularity Theorems

• Linear Pair Perpendicular Theorem: If two lines intersect to form a linear pair of congruent angles, the lines are perpendicular.
• Perpendicular Transversal Theorem: If a transversal is perpendicular to one of two parallel lines, it is also perpendicular to the other line.
• Lines Perpendicular to a Transversal Theorem: If two lines are perpendicular to the same line, those two lines are parallel.

Slopes of Lines

• Slopes of Parallel Lines: Two nonvertical lines in a coordinate plane are parallel if and only if their slopes are equal.
• Slopes of Perpendicular Lines: Two nonvertical lines in a coordinate plane are perpendicular if and only if the product of their slopes equals -1.