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# Chapter Three

### Parallel Lines and Transversals

Term | Definition |
---|---|

parallel lines | coplanar lines that do not intersect |

parallel planes | planes that do not intersect |

skew lines | lines that do not intersect and are not coplanar |

transversal | a line that intersects two or more coplanar lines at two different points |

same side (consecutive) interior angles | interior angles that lie on the same side of the transversal |

alternate interior angles | nonadjacent interior angles that line on opposite sides of the transversal |

alternate exterior angles | nonadjacent exterior angles that lie on opposite sides of the transversal |

corresponding angles | lie on the same side of the transversal and on the same side of the lines. |

Corresponding Angles Postulate | if two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent. |

Alternate Interior Angles Theorem | If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent. |

Consecutive Interior Angles Theorem | If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary. |

Alternate Exterior Angles Theorem | If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent. |

Perpendicular Transversal Theorem | In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other. |

slope | ratio of the change along the y-axis to the change along the x-axis between any two points on the line. |

Four Different Types of Slope | (positive slope, negative slope, zero slope, undefined slope) |

rate of change | how a quantity y changes in relationship to a quantity x. |

Slope of Parallel Lines Postulate | Two nonvertical lines have the same slope IFF they are parallel. Are vertical lines are parallel. |

Slope of Perpendicular Lines Postulate | Two nonvertical lines are perpendicular IFF the product of their slope is -1. Vertical and horizontal lines are perpendicular. |

Slope-Intercept Form | y=mx+b, where m is the slope of the line and b is the y intercept. |

point-slope form | y-y1=m(x-x1) where (x1,y1) is any point on the line and m is the slope of the line. |

Horizontal Line | The equation of a horizontal line is y=b where b is the y-intercept of the line. |

Vertical Line | The equation of a vertical line is x=a, where a is the x-intercept of the line. |

Postulate 3.4: Converse of Corresponding Angles Postulate | If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. |

Postulate 3.5 Parallel Postulate | If given a line and a point not on the line, then there exists exactly one line through the point that is parallel to the given line. |

Theorem 3.5 Alternate Exterior Angle Converse | If two lines in a plane are cut by a transversal so that a pair of alterate exterior angles is congruent, then the two lines are parallel. |

Theorem 3.6 Consecutive Interior Angles Converse | If two lines in a plane are cut by a transversal so that a pair of consecutive interior angles is supplementary, then the lines are parallel. |

Theorem 3.7: Alternate Interior Angles Converse | If two lines in a plane are cut by a transversal so that a pair of alternate interior angles is congruent, then the lines are parallel. |

Theorem 3.8: Perpendicular Transversal Converse | In a plane, if two lines are perpendicular to the same line, then they are parallell. |

Distance between a point and a line | The distance between a line and a point not on the line is the length of the segment perpendicular to the line from the point. |

Postulate 3.6 Perpendicular Postulate | If given a line and a point not on the line, then there exists exactly one line through the point that is perpendicular to the given line. |

equidistant | the distance between two lines measured along a perpendicular line to the lines is always the same. |

Distance between Parallel Lines | the distance betweeen 2 parallel lines is the perpendicular distance between one of the lines and any point on the other line. |

Theorem 3.9 Two Lines Equidistant from a Third | In a plane, if two lines are equidistant from a third line, then the two lines are parallel to each other. |

Created by:
amgeometry