Parallel Lines
Introduction
Parallel lines are lines in a plane that never cross or meet at any point. They always remain equidistant from each other. Parallel lines are like railroad tracks – they run side by side and never meet. They're always the same distance apart and have the same slant. When a line cuts across a set of parallel lines, it creates new angles. In this article, we will learn more about parallel lines, their properties, slope, etc with examples.
What Are Parallel Lines?
Parallel lines are lines that run side by side but never intersect. No matter how far you extend them, they'll never meet. Imagine two straight roads that run alongside each other but never merge – those are parallel lines.
Examples of Parallel Lines
You can find parallel lines all around you, even in everyday objects. Think about the sides of a square – they're like two parallel lines that never touch. When you look at a table, the edges where the surface meets the legs form parallel lines. These lines help us understand shapes and objects in our world.
Parallel Lines Examples in Real Life
- Railway tracks
- Power lines on poles
- Lines on notebooks
- Window blinds
- Zebra crossings
- Shelves in a bookcase
Parallel Lines Symbol
When lines never meet no matter how far they stretch, they're called parallel lines. To represent this in math, we use a special symbol: `||`. So, if we see parallel symbol like `AB` `||` `CD`, it means line `AB` and line `CD` are parallel. On the other hand, if lines do meet, they're not parallel. We show this with a different symbol: `∦`. For instance, if we have `EF ∦ GH` it means `EF` and `GH` are not parallel.
Parallel Lines and Transversal
When two parallel lines are crossed by another line, known as a transversal, a fascinating world of angles is revealed. Imagine two roads running side by side, with a third road intersecting them at different points – that's what happens when parallel lines meet a transversal. In the world of geometry, this intersection creates a various types of angles, each with its unique properties.
In the figure below, we see two parallel lines, labeled `L_1` and `L_2`, intersected by a transversal `t`. As a result, eight distinct angles emerge, each denoted by a letter. Let's explore the relationships between these angles:
- Corresponding Angles: Angles in matching positions are equal. For instance, `∠a = ∠e, ∠b =∠f, ∠c =∠g` and `∠d =∠h`.
- Alternate Interior Angles: These angles are on the inner sides of the parallel lines, but on opposite sides of the transversal. They share the same measure. In our diagram, `∠c` and `∠e`, and `∠d` and `∠f` form pairs of alternate interior angles.Hence, `∠c = ∠e`, and `∠d = ∠f`.
- Alternate Exterior Angles: These angles are on opposite sides of the transversal but outside the parallel lines. They too have equal measures. In this example, `∠a` and `∠g`, and `∠b` and `∠h` form pairs of alternate interior angles. Hence, `∠a = ∠g`, and `∠b = ∠h`.
- Consecutive Interior Angles: Also known as co-interior angles, they are inside the parallel lines and on the same side of the transversal. The sum of consecutive interior angles is `180` degrees. In this example `∠c` and `∠f`, and `∠d` and `∠e` form pairs of co-interior angles. Thus, `∠c + ∠f = 180` degrees, as does `∠d + ∠e`.
- Vertically Opposite Angles: When two lines intersect, the angles opposite each other are equal. Therefore, `∠a = ∠c, ∠b = ∠d, ∠e = ∠g`, and `∠f = ∠h`. Thus `∠a` and `∠c, ∠b` and `∠d, ∠e` and `∠g, ∠f` and `∠h` form pairs on vertically opposite angles.
Parallel Line Equations
Linear equations, which describe straight lines, are commonly represented by slope-intercept form, given by the equation ` y = mx + b `. Here, `m` represents the slope, `b` is the `y`-intercept, and `y` and `x` are variables. The slope `m` determines how steep the line is.
Interestingly, for parallel lines, their slopes are always identical. For instance, if we have a line with the equation ` y = 3x + 2 `, and its slope is `3`, then any line parallel to it will also have a slope of `3`. However, despite sharing the same slope, parallel lines have different `y`-intercepts, meaning they intersect the `y`-axis at different points and never touch or share any points or angles.
The figure below illustrates two parallel lines, each having a slope of `-2`.
Solved Examples
Example `1`: Determine whether the lines in the figure are parallel or not.
Solution: The given lines in the figure are intersected by a transversal. To determine if they are parallel, we need to check if any pairs of alternate interior angles are equal.
Alternate Angles `a` and `b`:
`∠a` and `∠b` are alternate interior angles.
Since `∠a = 60°` and `∠b = 60°`, they are equal.
Therefore, lines `a` and `b` are parallel.
Example `2`: In the diagram given below, `AB` `||` `CD`. Identify a pair of consecutive interior angles and a pair of corresponding angles in the following diagram.
Solution:
In the given diagram, two parallel lines are intersected by a transversal.
`∠3` and `∠5` form a pair of consecutive interior angles.
`∠2` and `∠6` make a pair of corresponding angles.
Example `3`: Are the lines represented by the equations `y = -\frac{2}{3}x + 4` and `6x + 9y = 30` parallel?
Solution:
The equation for the first line `y = -\frac{2}{3}x + 4` is given in the slope-intercept form `y = mx + b`. Hence slope of the first line is `-\frac{2}{3}`.
The equation of the second line `6x + 9y = 30` is written in standard form.
We can solve the equation for `y` to write it in slope-intercept form.
\( 6x + 9y = 30 \)
\( 9y = -6x + 30 \)
`y = -\frac{6}{9}x + \frac{30}{9}`
Hence the slope of the line is `-\frac{6}{9}`, which can be simplified to `-\frac{2}{3}`.
Since the slope of the second line is the same as the slope of the first line, both lines are parallel.
Example `4`: Find the equation of a line parallel to `y = 5x + 8` and passes through the point `(2,6)`.
Solution:
The line parallel to `y = 5x + 8` will have the same slope as the slope of `y = 5x + 8`, which is `5`.
As the parallel line passes through `(2,6)`, we can use this point to calculate the `y`-intercept of the parallel line.
\( y = 5x + b \)
Plugging in `x = 2` and `y = 6`, we get
\( 6 = 5(2) + b \)
\( b = -4 \)
Hence the parallel line has a slope of `5` and `y`-intercept at `(0,-4)`. The equation of the parallel line is `y = 5x -4`.
Example `5`: Find the value of `x` and `y` in the given figure where `L_1` is parallel to `L_2`.
Solution:
As `L_1` is parallel to `L_2`, the pair of alternate exterior angles `(4x)` and `(x + y)` are congruent. Hence
\( 4x = x + y \)
\( y = 3x \)
Also `(4x)` and `(x + 5y)` form a a pair of supplementary angles. Supplementary angles add to `180°`. Hence
\( 4x + x + 5y = 180 \)
Substituting `y = 3x` we get
\( 4x + x + 5(3x) = 180 \)
\( 4x + x + 15x = 180 \)
\( 20x = 180 \)
\( x = 9 \)
We know `y = 3x`. Plugging `9` for `x`, we get `y = 27`.
Therefore, the values for `x` and `y` are `9` and `27` respectively.
Practice Problems
Q`1`. Which of the given lines look parallel?
Answer: b
Q`2`. Which shape has a pair of lines that are parallel?
Answer: d
Q`3`. What is the measure of the unknown angle if the lines '`p`' and '`q`' are parallel?
- `78°`
- `82°`
- `92°`
- `102°`
Answer: d
Q`4`. What will be the slope of a line parallel to `2x - 3y = 7`?
- `\frac{3}{2}`
- `\frac{2}{3}`
- `-\frac{2}{3}`
- `-\frac{3}{2}`
Answer: b
Q`5`. What is the equation of a line parallel to `y = -4x + 6` passing through the point `(-2, 6)`?
- `y = 4x + 6`
- `y = -4x - 6`
- `y = -\frac{1}{4} + 10`
- `y = -4x - 2`
Answer: d
Frequently Asked Questions
Q`1`. What are parallel lines?
Answer: Parallel lines are two or more lines in the same plane that never intersect each other, no matter how far they extend.
Q`2`. How do you identify parallel lines?
Answer: Parallel lines can be identified by checking if they have the same slope and never intersect, even when extended infinitely.
Q`3`. What are the properties of parallel lines?
Answer: The properties of parallel lines include being equidistant from each other, having the same slope, and forming congruent corresponding angles when intersecting by a transversal.
Q`4`. What is the symbol for parallel lines?
Answer: The symbol used to denote parallel lines is '`||`'.
Q`5`. Can parallel lines exist in three-dimensional space?
Answer: Yes, parallel lines can exist in three-dimensional space as long as they are in the same plane and never intersect, similar to how they exist in two-dimensional space.