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Larson Geometry Reference – Chapter 3 Definitions Term/Concept Description 1 & 7 2 &8 *Alternate Exterior Angles 1 2 8 7 4 & 6 3 &5 *Alternate Interior Angles 4 6 4 & 5 3 & 6 *Consecutive Angles (Consecutive Interior Angles) 4 3 5 1 & 5 4 &8 2 &6 3 &7 *Corresponding Angles 3 5 6 1 2 4 5 8 given a segment AB, start at point A and move to point B along A proof written in paragraph form. Coplanar lines that never intersect. Lines that intersect at right angles. Two planes that never intersect. Non-coplanar lines that never intersect. The ratio of the vertical change (rise) to horizontal change (run) between any two points on a line. The slope m of a line containing two points with coordinates (x 1, y1) and (x2, y2) is given Paragraph Proof *Parallel Lines *Perpendicular Lines Parallel Planes *Skew lines *Slope of line by the formula: m y 2 y1 , where x1 x2. x 2 x1 A line that intersects two or more coplanar lines. *Transversal k Lines m and n are parallel. (m ∥ n.) Lines m and k are skew lines. m U 7 The distance from a line to a point not on the line is the length of the segment perpendicular to the line from the point. A segment that represents moving a specific distance in a specific direction. It is also a quantity that has length and direction. Also known as a vector. For example, in a coordinate plane Distance Between a Point and a Line Directed Line Segment T 3 6 n Planes T and U are parallel planes. (T ∥ U) Lines k and n are intersecting lines. Larson Geometry Reference – Chapter 3 Postulates Postulate Corresponding Angles Corresponding Angles Converse Parallel Postulate Perpendicular Postulate Slope of Parallel Lines Slopes of Perpendicular Lines If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent If two lines are cut by a transversal so that the corresponding angles is congruent, then the lines are parallel. If there is 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 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. In a coordinate plane, two non-vertical lines are parallel if and only if they have the same slope. In a coordinate plane, two non-vertical lines are perpendicular if and only if the product of their slopes is –1. That means that the slopes are negative reciprocals of each other. Theorems Theorems *Alternate Interior Angles Alternate Interior Angle Converse *Alternate Exterior Angles Alternate Exterior Angle Converse Complementary Adjacent Angles If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent. If two lines in a plane are cut by a transversal so that a pair of alternate interior angles is congruent, then the two lines are parallel. If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent If two lines in a plane are cut by a transversal so that a pair of alternate exterior angles is congruent, then the two lines are parallel. If two sides of two adjacent acute angles are perpendicular, then the angles are complementary. A If BA BC , then 1 and 2 are complementary. 1 2 B *Consecutive Interior Angles Consecutive Interior Angles Converse *Corresponding Angles Corresponding Angles Converse Four Right Angles Linear Pair Perpendicular Lines Perpendicular to a Transversal Parallel Line Transitivity Perpendicular Transversal Theorem C If two parallel lines are cut by a transversal, then each pair of consecutive interior angles are supplementary. If two lines in a plane are cut by a transversal so that a pair of consecutive interior angles are supplementary, then the two lines are parallel. If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent If two lines are cut by a transversal so that the corresponding angles is congruent, then the lines are parallel. If two lines are perpendicular, then they intersect to form four right angles. If two lines intersect to form a linear pair of congruent angles then the lines are perpendicular. In a plane, if two lines are perpendicular to the same line, then they are parallel If two lines are parallel to the same line, then they are parallel to each other. In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.