Chapter 3 Parallel and Perpendicular Lines

3.1 Identify Pairs of Lines and Angles  Parallel lines- ( II ) do not intersect and are coplanar  Parallel planes- do not intersect  Skew lines- do not intersect and are NOT coplanar

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.

 Transversal- a line that intersects two or more coplanar lines at different points

 Corresponding angles- 2 angles in the same corresponding position relative to the 2 lines  Alternate interior angles- 2 nonadjacent interior angles on opposite sides of the transversal  Alternate exterior angles- 2 exterior angles on opposite sides of the transversal  Consecutive interior angles (same side interior)- 2 interior angles on the same side of the transversal

 Corresponding  Alternate interior  Alternate exterior  Consecutive interior

Example

3.2 Using Parallel Lines and Transversals  Corresponding Angles Postulate- If 2 parallel lines are cut by a transversal, then corresponding angles are congruent.

 Alternate Interior Angles Theorem- If 2 parallel lines are cut by a transversal, then alternate interior angles are congruent.  Alternate Exterior Angles Theorem- If 2 parallel lines are cut by a transversal, then alternate exterior angles are congruent.

 Consecutive Interior Angles Theorem- If 2 parallel lines are cut by a transversal, then consecutive interior angles are supplementary.

 Perpendicular Transversal Theorem- If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other one also.

Prove the Alternate Interior Angles Theorem

Example

3.3 Proving Lines are Parallel  Corresponding Angles Converse (Postulate)- If two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel.

Theorems  Alternate Interior Angles Converse - If two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel.  Alternate Exterior Angles Converse - If two lines are cut by a transversal so the alternate exterior angles are congruent, then the lines are parallel.

 Consecutive Interior Angles Converse - If two lines are cut by a transversal so the consecutive interior angles are supplementary, then the lines are parallel.

 Transitive Property of Parallel Lines- If two lines are parallel to the same line, then they are parallel to each other.

2-column proof of Alternate Interior Angles Converse

3.4 Find and Use Slopes of Lines  Slope, m = or  Steeper slopes have higher absolute values.

Example  Line k passes through (0,3) and (5,2). Graph the line perpendicular to k and passes through the point (1,2). Find the equation of the perpendicular line.

Example  Graph using the x and y intercept: 3x + 4y = 12

Example  Your gym membership has a one time initial fee and a monthly fee.  2 months total spent = $ 231  5months total spent = $ 363  Write an equation in slope intercept form to represent the cost of the membership.  What does the slope represent?  What does the y intercept represent?

3.5 Write and Graph Equations of Lines  Write the equation of the line in slope intercept form.

Example  Write the equation of the line passing through ( -1, 1) and parallel to y = 2x – 3.

Example  Write the equation of the line passing through ( 2, 3) and perpendicular to y = ½ x + 2.

Example  Write the equation of the line that is the perpendicular bisector of the line segment with endpoints: P( -4, 3 ) Q (4, -1).

3.6 Prove Theorems about Perpendicular Lines  If 2 lines are perpendicular, then they intersect to form 4 right angles.  If 2 lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.  If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.

Find the distance between a point and a line  Line m contains (0, -3) and (7, 4). P (4, 3)

Find the distance between 2 parallel lines  Line j: y = 5x – 22  Line m: y = 5x + 4