Slope and Parallel Lines Sections 4.5 & 4.6

Definitions A plane is a surface such that if any two points on the surface are connected by a line, all points of the line are also on the surface. A plane has only two dimensions – length and width – but no thickness.

Definitions If points, lines, segments, and so forth lie in the same plane, we call them coplanar. Points, lines, segments, and so forth that do not lie in the same plane are called noncoplanar.

Definitions A transversal is a line that intersects two coplanar lines in two distinct points.

Definitions In the diagram, the region between lines d and e is the interior of the figure. In the diagram, the rest of the plane except the region between lines d and e is the exterior of the figure.

Definitions Alternate Interior Angles are a pair of angles formed by two lines and a transversal. The angles must –both lie in the interior of the figure, –lie on alternate sides of the transversal, –have different vertices.

Definitions Alternate Exterior Angles are a pair of angles formed by two lines and a transversal. The angles must –both lie in the exterior of the figure, –lie on alternate sides of the transversal, –have different vertices.

Definitions Corresponding Angles are a pair of angles formed by two lines and a transversal. –One angle must lie in the interior of the figure, and the other must lie in the exterior. –The angles must lie on the same side of the transversal but have different vertices.

Parallel Lines Parallel (║) lines are two coplanar lines which do not intersect. Parallel lines have the same slope.

Slope Review The slope of a nonvertical line (or segment or ray) containing points (x 1, y 1 ) and (x 2, y 2 ) is defined by Find the slope of the line containing points (2, -1) and (7, 4)

Remember, Rising line – positive slope Falling line – negative slope Horizontal line – zero slope Vertical line – no slope (undefined slope)

Slopes of Parallel Lines Theorem 26: If two nonvertical lines are parallel, then their slopes are equal. Theorem 27: If the slopes of two nonvertical lines are equal, then the lines are parallel.

Slopes of Perpendicular Lines Theorem 28: If two nonvertical lines are perpendicular, then each line’s slope is the opposite reciprocal of the other’s. Theorem 29: If a line’s slope is the opposite reciprocal of another line’s slope, then the two lines are perpendicular. Flip the top and bottom of fraction and change to opposite sign!