|
If two lines in a plane are cut by a transversal and certain conditions are met, then the lines must be parallel:
- corresponding angles are congruent
- alternate exterior angles are congruent
- consecutive interior angles are supplementary
- alternate interior angles are congruent
- two lines are perpendicular to the same line
You can prove that lines are parallel by using postulates and theorems about pairs of angles. You can also use slopes of lines to prove that two lines are parallel or perpendicular.
 Converse of the Corresponding Angles Postulate
If two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel.
The Converse of the Corresponding Angles Postulate is used to construct parallel lines.
The Parallel Postulate guarantees that for any line m, you can always construct a parallel line through a point that is not on line m.
 Converse of the Alternate Interior Angles Theorem
If two coplanar lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the two lines are parallel.
Converse of the Alternate Exterior Angles Theorem
If two coplanar lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then the two lines are parallel.
Converse of the Same-Side Interior Angles Theorem
If two coplanar lines are cut by a transversal so that a pair of same-side interior angles are supplementary, then the two lines are parallel.
Write Equations of Lines
A linear relationship between two variables can be represented by an equation in point-slope form or slope-intercept form. The equation can then be used to analyze the relationship.
You can write an equation of a line if you are given any of the following:
- the slope and the y-intercept
- the slope and the coordinates of a point on the line
- the coordinates of two points on the line
If m is the slope of the line, b is its y-intercept, and (x1, y1) is a point on the line, then:
- the slope-intercept form of the equation is y = mx + b
- the point-slope form of the equation is y - y1 = m(x - x1)
Parallel Lines Theorem
In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel.
Perpendicular Lines Theorem
In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1 (they are opposite reciprocals of each other). Vertical and horizontal lines are perpendicular.
This video demonstrates how to write the equation of a line, a line...
This video demonstrates how to write the equation of a line, a line parallel to the original, and a line perpendicular to the original.
Classifying Pairs of Lines
In a system of equations that represents two lines, you need to determine whether the lines are parallel, intersect, or coincide. Lines that are parallel have the same slope, but different y-intercepts. Lines that have different slopes, will intersect each other. Lines that coincide are the same line, but their equations may be written in different forms. In order to determine which classification applies, you need to solve both equations for y to find the slope-intercept form: y = mx + b.
Determine whether the lines are parallel, intersect, or coincide. ...
Determine whether the lines are parallel, intersect, or coincide. Solve both equations for y to find the slope-intercept form of the equations. Compare the slopes and y-intercepts.
Visit www.msdgeometry.com - your ultimate internet resource for comprehensive information on High School Geometry, Algebra, and Trigonometry.
|
Chapter 3
When two lines lie in the same plane and do not intersect, they are parallel to each other.
Alternate Interior Angles
Alternate Exterior Angles
Same-Side Interior Angles
