Two or more lines that lie in the same plane and never intersect each other are known as parallel lines. They are equidistant from each other and have the same slope. Let us learn more about parallel lines in this article.
|1.||What are Parallel Lines?|
|2.||Parallel Lines and Transversal|
|3.||Parallel Lines Properties|
|4.||FAQs on Parallel Lines|
What are Parallel Lines?
Parallel lines are straight lines that never meet each other no matter how long we extend them. Observe the following figure that shows parallel lines. Line 'a' is parallel to line 'b', and line 'p' is parallel to line 'q'.
Parallel Lines and Transversal
When any two parallel lines are intersected by another line called a transversal, many pairs of angles are formed. While some angles are congruent (equal), the others are supplementary. Observe the following figure to see the parallel lines labeled as L1 and L2 that are cut by a transversal. Eight separate angles have been formed by the two parallel lines and a transversal. Each angle has been labeled using an alphabet.
Given below are the pairs of angles formed by the two parallel lines L1 and L2.
- Corresponding Angles: It should be noted that the pair of corresponding angles are equal in measure. In the given figure, there are four pairs of corresponding angles, that is, ∠a = ∠e, ∠b = ∠f, ∠c = ∠g, and ∠d = ∠h
- Alternate Interior Angles: Alternate interior angles are formed on the inside of two parallel lines that are intersected by a transversal. They are equal in measure. In this figure, ∠c = ∠e, ∠d = ∠f
- Alternate Exterior Angles: Alternate exterior angles are formed on either side of the transversal and they are equal in measure. In this figure, ∠a = ∠g, ∠b = ∠h
- Consecutive Interior Angles: Consecutive interior angles or co-interior angles are formed on the inside of the transversal and they are supplementary. Here, ∠c + ∠f = 180°, and ∠d + ∠e = 180°
- Vertically Opposite Angles: Vertically opposite angles are formed when two straight lines intersect each other and they are equal in measure. Here, ∠a = ∠c, ∠b = ∠d, ∠e = ∠g, ∠f = ∠h
Parallel Lines Properties
Parallel Lines can be easily identified with the basic properties given below.
- Parallel lines are those straight lines that are always the same distance apart from each other.
- Parallel lines never meet no matter how much they are extended in either direction.
How do you Know if Lines are Parallel?
Apart from the characteristics given above, when any two parallel lines are cut by a transversal, they can be identified by the following properties.
- Any two lines are said to be parallel if the Corresponding angles so formed are equal.
- Any two lines are said to be parallel if the Alternate interior angles so formed are equal.
- Any two lines are said to be parallel if the Alternate exterior angles so formed are equal.
- Any two lines are said to be parallel if the Consecutive interior angles on the same side of the transversal are supplementary.
Parallel Lines Equation
The equation of a straight line is generally written in the slope-intercept form represented by the equation, y = mx + b, where 'm' is the slope and 'b' is the y-intercept. The value of 'm' determines the slope or gradient and tells us how steep the line is.
It should be noted that the slope of any two parallel lines is always the same. For example, if the slope of a line with the equation y = 4x + 3 is 4. Therefore, any line that is parallel to y = 4x + 3 will have the same slope, that is, 4. Parallel lines have different y-intercepts and have no points in common.
Parallel Lines Symbol
Parallel lines are the lines that never meet each other, no matter how long we extend them. The symbol used to denote parallel lines is ||. For example, AB II PQ indicates that line AB is parallel to line PQ. The symbol that denotes non-parallel lines is ∦.
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Parallel Lines Examples
Example 1: Using the properties of parallel lines, write true or false for the following statements.
a.) Parallel lines always intersect at 90°.
b.) Parallel lines are always the same distance apart.
a.) False, parallel lines never intersect each other. However, perpendicular lines always intersect at 90°.
b.) True, parallel lines are always the same distance apart.
Example 2: Name any two pairs of angles that are formed when any two parallel lines are cut by a transversal.
When any two parallel lines are cut by a transversal, many pairs of angles are formed. Two of them are Corresponding angles and Alternate interior angles.
Example 3: Identify one pair of Alternate Interior angles and a pair of corresponding angles in the following figure.
Solution: In the given figure, two parallel lines are cut by a transversal. Therefore, ∠2 and ∠3 form a pair of Alternate Interior angles, which means ∠2 = ∠3. And ∠2 and ∠5 form a pair of corresponding angles, which means, ∠2 = ∠5
FAQs on Parallel Lines
What are Parallel Lines in Geometry?
Parallel lines are those lines that are always the same distance apart and that never meet. The symbol used to denote parallel lines is ||. For example, AB||CD means line AB is parallel to line CD.
What are Parallel Lines and Perpendicular lines?
Parallel lines are those lines that are equidistant from each other and never meet, no matter how much they may be extended in either directions. For example, the opposite sides of a rectangle represent parallel lines. On the other hand, if any two lines intersect each other at 90°, they are called perpendicular lines. For example, the adjacent sides of a rectangle are perpendicular lines because they intersect each other at 90°.
What are the Types of Angles in Parallel Lines?
When any two parallel lines are intersected by a transversal, they form many pairs of angles, like, Corresponding angles, Alternate interior angles, Alternate exterior angles, and Consecutive interior angles.
What Happens when Parallel Lines are Cut by a Transversal?
When any two parallel lines get intersected by a transversal, the following angles are formed.
- Corresponding angles, that are equal in measure.
- Alternate Interior angles, that are equal in measure.
- Alternate Exterior angles, that are equal in measure.
- Consecutive Interior angles, that are supplementary.
What do Parallel Lines Look Like?
Parallel lines look like railway tracks that never meet and are always equidistant. The opposite sides of a rectangle also represent parallel lines that are equidistant.
What is the Slope of Parallel Lines?
If two lines are parallel, they have the same slope. For example, if the equation of a straight line is y = 1/2x + 17, its slope is 1/2. Now, if a line has the same slope of 1/2 in the same plane, it will be parallel to the given line.
What are Real-World Examples of Parallel Lines?
The real-life examples of parallel lines include railroad tracks, the edges of sidewalks, rails of a ladder, never-ending rail tracks, opposite sides of a ruler, opposite edges of a pen, eraser, etc.
What is the Rule for Parallel Lines?
The rule for parallel lines is that the lines should not meet each other. In other words, if two straight lines in the same plane are the same distance apart and they never meet each other, they are called parallel lines.
What is the Symbol for Parallel Lines?
The symbol that is used to denote parallel lines is ||. Any two parallel lines AB and CD are represented as AB||CD.
Do Parallel Lines Meet at Infinity?
No, the definition itself suggests that parallel lines never meet. Hence, parallel lines would not meet even at infinity.
Do Parallel Lines have the Same Equation?
No, parallel lines do not have the same equation, but they have the same slope. For example, if the equation of a line is represented as, y = 4x + 2, this means the slope of this line is 4. So, another straight line in the same plane, that has the same slope of 4 will be parallel to the given line.
Are Parallel Lines Equal in Length?
No, parallel lines may not be equal in length but they should be the same distance apart.
Does a Triangle have Parallel Lines?
No, a triangle does not have any parallel lines. Since a triangle always has 3 intersecting sides; and we know that parallel lines never intersect each other, therefore, a triangle cannot have parallel lines.
How many Parallel Lines does a Hexagon have?
A hexagon is a six-sided polygon. A regular hexagon has three pairs of parallel lines.