Parallel lines
Parallel lines are lines in the same plane that do not intersect. Line segments and rays that are parts of parallel lines are also parallel.
The symbol to show parallel lines is "//". In the figure below, m//n.
You can also place arrows on the lines, m and n, as in the figure above, to show they are parallel.
Transversals of parallel lines and their angles
When 2 lines are cut (intersected) by a third line, called a transversal, 8 angles are formed.
Parallel lines m and n are cut by transversal l, above, forming angles 1–8.
- ∠1, ∠2, ∠7, ∠8 are called exterior angles.
- ∠3, ∠4, ∠5, ∠6 are called interior angles.
- ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8 are called corresponding angles.
- ∠3 and ∠6, ∠4 and ∠5 are pairs of consecutive interior angles.
- ∠1 and ∠8, ∠2 and ∠7 are pairs of exterior angles on the same side of the transversal.
Several relationships exist among these angles.
- Alternating exterior angles are congruent. So, ∠1≅∠7 and ∠2≅∠8.
- Alternate interior angles are congruent. So, ∠3≅∠5 and ∠4≅∠6.
- Corresponding angles on the same side of the transversal are congruent. So, ∠1≅∠5, ∠2≅∠6, ∠3≅∠7, and ∠4≅∠8.
- Consecutive interior angles are supplementary. So, ∠3 + ∠6 = 180° and ∠4 + ∠5 = 180°.
- Exterior angles on the same side of the transversal are supplementary. So, ∠1 + ∠8 = 180° and ∠2 + ∠7 = 180°
In summary, ∠1, ∠3, ∠5, ∠7 are congruent and ∠2, ∠4, ∠6, ∠8 are also congruent. ∠1, ∠3, ∠5, ∠7 are supplementary to ∠2, ∠4, ∠6, ∠8, respectively.
If any of the eight angles formed by two parallel lines and a transversal is a right angle, all the angles formed are right angles and the transversal is perpendicular to the two parallel lines.
Slope of parallel lines
In coordinate geometry, parallel lines have the same slope. The converse is also true; if two lines have the same slope, the two lines are parallel unless they overlap.
The blue line below is the graph of the equation y = 2x + 3 and the black line is y = 2x - 4. The slope for both lines is, m = 2. Two vertical lines are still parallel even though their slopes are undefined.
Knowing that two lines are parallel is useful for finding the equation of a line even when given little information about it.
Example:
Suppose a line contains the point (5, -2) and is parallel to the line that has an equation of y = 2x + 6.
The slope of the given line is 2.
Using the point-slope formula, we have,
| y - (-2) = 2(x - 5) |
| y = 2x - 12 |
See also intersecting lines, perpendicular lines.