CHAPTER 3 Parallel & Perpendicular lines
3.1- Identify Pairs of Lines and Angles Parallel Lines: two lines are parallel if they do not intersect and are coplaner Skew lines: Two lines are skew if they do not intersect and are not coplaner Parallel planes: Two planes that do not intersect
Parallel & Perpendicular Lines Two lines in same plane are either parallel or intersect Through a point not on a line, there are infinitely many lines Exactly one is parallel to the given line Exactly one is perpendicular to the given line
Parallel & Perpendicular Postulates Two of Euclid s most important postulates are the parallel and perpendicular postulates 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 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.
Angles and Transversals Transversal A line that intersects two or more coplaner lines at different points Angles Formed by Transversals Corresponding Angles Two angles that have corresponding postions Alternate Interior Angles Two angles that lie between the two lines and on opposite sides of the transversal Alternate Exterior Angles Two angles that lie outside the two lines and on opposite sides of the transversal Consecutive Interior Angles (Same-Side Interior) Two angles that lie between the two lines and on the same side of the transversal
3.2- Use Parallel Lines & Transversals Several postulates and theorems exist about parallel lines and transversals, which help to prove angles congruent Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent
Theorems Theorem 3.1 Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent Theorem 3.2 Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent Theorem 3.3 Consecutive Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles (same-side interior) are supplementary (add up to 180 )
3.3 Prove Lines are Parallel The converse of the Corresponding Angles Postulate, as well as the converse of the theorems from section 3.2 exist The converse of a true conditional statement is not necessarily true Therefore each converse must be proved
Corresponding Angles Corresponding Angles Postulate Converse If two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel
Theorems Converse Theorem 3.4 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 3.5 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 3.6 Consecutive Interior Angles Converse If two lines are cut by a transversal so the consecutive (same-side) interior angles are supplementary, then the lines are parallel
Transitive Property of Parallel Lines The transitive property also applies to parallel lines Theorem 3.7 Transitive Property of Parallel Lines If two lines are parallel to the same line, then they are parallel to each other
3.4 Find and Use Slopes of Lines Slope Ratio of vertical change (rise) to horizontal change (run) between any two points on the line Slope of Lines in the Coordinate Plane Negative Slope falls from left to right Positive Slope rises from left to right Zero Slope horizontal line Undefined Slope vertical line The slope of a line will be used to solve problems involving parallel and perpendicular lines
Comparing Slopes When two lines intersect, the steeper line has a slope with greater absolute value Slopes of parallel and perpendicular lines can also be compared Slopes of Parallel Lines Postulate In a coordinate plane, two non-vertical lines are parallel if and only if they have the same slope Slopes of Perpendicular Lines Postulate In a coordinate plane, two non-vertical lines are perpendicular if and only if their slopes are opposite reciprocals (product is -1) Horizontal lines are perpendicular to vertical lines
3.5 Write and Graph Equations of Lines Linear equations may be written in different forms Slope-Intercept Form y = mx + b where m is the slope and b is the y-intercept Standard Form Ax + By = C where A and B are not both zero
3.6 Prove Theorems About Perp. Lines Theorem 3.8 If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular Theorem 3.9 If two lines are perpendicular, then they intersect to form four right angles Theorem 3.10 If two sides of two adjacent acute angles are perpendicular, then the angles are complementary
Theorems continued Theorem 3.11 Perpendicular Transversal Theorem If a transversal perpendicular to one of two parallel lines, then it is perpendicular to the other Theorem 3.12 Lines Perpendicular to a Transversal In a plane, if two lines are perpendicular to the same line, then they are parallel to each other
Distance From a Point to a Line The distance from a point to a line is the length of the perpendicular segment from the point to the line The perpendicular segment is the shortest distance between the point and the line The distance between two parallel lines is the length of any perpendicular segment joining the lines