notes unit 4 parallel and perpendicular lines distance and midpoint equations for lines
TRANSCRIPT
Notes Unit 4
Parallel and Perpendicular Lines
Distance and MidpointEquations for Lines
Definition of Parallel Lines (//)
Two lines that lie in the same plane that never intersect are called parallel.Lines m & n are parallel
Definition of Skew Lines
Two lines are skew if they do not intersect and do not lie in the same plane. Lines m & k are skew
Definition of Parallel Planes
Two planes that do not intersect.
Planes T & U are parallel
Definition of Perpendicular Lines
Perpendicular lines are lines that intersect to form a right angle.Line CD and Line DE are perpendicular
Definition of Perpendicular Planes
Planes that intersect to form a right angle.Planes ABC and ABG are perpendicular.
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. There is exactly one line through P parallel to line l.
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. There is exactly one line through P perpendicularto line l.
Corresponding Angles postulate
• Two lines cut by a transversal are parallel if and only if the pairs of corresponding angles are congruent.
Alternate Interior Angles Theorem
• Two lines cut by a transversal are parallel if and only if the pairs of alternate interior angles are congruent.
Alternate exterior angles theorem
• Two lines but by a transversal are parallel if and only if the pairs of alternate exterior angles are congruent.
Consecutive Interior Angles Theorem
• Two lines cut by a transversal are parallel if and only if the pairs of consecutive interior angles are supplementary.
Example
• Find the value of x.
Example
• Find the value of x. The picture may not be drawn to scale.
(3x + 5)o
(7x – 15)o
Transitive Property of Parallel Lines
If two lines are // to the same line, then they are // to each other.
Perpendicular Transversal Theorem
If a transversal is to one of two // lines, then it is to the other.
If line j line h and line h and line k are //, then line j line k
Lines Perpendicular to a Transversal Theorem
In a plane, if 2 lines are to the same line, then they are // to each other.
If lines m & n are both to line p, then lines m & n are //.
Slope
the change in y divided by the change in x
Formula: Slope = y2 – y1
x2 – x1
Postulate – Slope of Parallel Lines
In the same plane, // lines have = slopes.
Postulate – Slope of Perpendicular Lines
In the same plane, lines have slopes that are negative reciprocals of each other.
Definition – Distance from a point to a Line
The distance between a point and a line must be measured with a segment from the point to the line.
Example• Graph the line y = x + 1. What point on the
line is the shortest distance from the point (4, 1)? What is the distance?