notes unit 4 parallel and perpendicular lines distance and midpoint equations for lines

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?