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In geometry, you study different types of lines such as intersecting lines, non-intersecting lines, concurrent lines, perpendicular lines, half lines, etc. Parallel lines are one such type of line in geometry having applications in many other topics.
Let’s understand what are parallel lines in geometry and their properties with examples.
What are Parallel Lines in Geometry?
Two lines are said to be parallel when they do not meet at any point in a plane. Parallel lines are lines that do not have a common intersection point and never cross paths with each other. The symbol for showing parallel lines is ‘||’.
Two lines that are parallel are represented as $\overleftrightarrow{\text{AB}} || \overleftrightarrow{\text{CD}}$
This means that line AB is parallel to CD.
The perpendicular distance between the two parallel lines is always constant.
In the above figure, the lines $\overleftrightarrow{\text{AB}}$ and $\overleftrightarrow{\text{CD}}$ represent two parallel lines as they have no common intersection point in the given plane. We can draw infinite parallel lines to lines $\overleftrightarrow{\text{AB}}$ and $\overleftrightarrow{\text{CD}}$ in the given plane.
Properties of Parallel Lines
The following are the basic properties of parallel lines that help to identify them.
- 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.
What is a Transversal?
A transversal is defined as a line that passes through two lines in the same plane at two distinct points in the geometry. A transversal intersection with two lines produces various types of angles in pairs, such as consecutive interior angles, corresponding angles, and alternate angles.
For a pair of lines (parallel lines or intersecting lines) a transversal produces eight angles as drawn in the figure below.
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.
Consider the following parallel lines $l_1$ and $l_2$ cut by a transversal line $t$. In such a case eight angles are formed by the two parallel lines and a transversal.
In the above figure, the following pairs of angles are present
- Vertically opposite angles: Vertically opposite angles are formed when two straight lines intersect each other and they are equal in measure. Here, the pair of vertically opposite angles are $\angle 1$ & $\angle 3$, $\angle 2$ & $\angle 4$, $\angle 5$ & $\angle 7$, and $\angle 6$ & $\angle 8$
- Corresponding angles: Corresponding angles are formed on the same side of the transversal. Here, the pair of corresponding angles are $\angle 1$ & $\angle 5$, $\angle 2$ & $\angle 6$, $\angle 3$ & $\angle 7$, and $\angle 4$ & $\angle 8$
- Interior angles on the same side of a transversal: Interior angles on the same side of the transversal or co-interior angles are formed on the inside of the transversal. Here, the pair of interior angles on the same side of the transversal are $\angle 3$ & $\angle 6$, $\angle 4$ & $\angle 5$
- Interior alternate angles: Interior alternate interior angles are formed on the inside of two parallel lines that are intersected by a transversal. Here, the pair of interior alternate angles are $\angle 3$ & $\angle 5$, $\angle 4$ & $\angle 6$
- Alternate exterior angles: Alternate exterior angles are formed on either side of the transversal. Here, the pair of alternate exterior angles are $\angle 1$ & $\angle 7$, $\angle 2$ & $\angle 8$
Properties of Parallel Lines and Transversal
If a transversal intersects two parallel lines, at two distinct points, then there are four angles formed at each point. Following are the properties of parallel lines with respect to transversals.
- Corresponding angles are equal.
- Vertically opposite angles are equal.
- Alternate interior angles are equal.
- Alternate exterior angles are equal.
- Pair of interior angles on the same side of the transversal are supplementary.
Let’s again consider the above figure.
In the above figure,
- $\angle 1$ & $\angle 3$, $\angle 2$ & $\angle 4$, $\angle 5$ & $\angle 7$, and $\angle 6$ & $\angle 8$ are vertically opposite angles, therefore, $\angle 1 = \angle 3$, $\angle 2 = \angle 4$, $\angle 5 = \angle 7$, and $\angle 6 = \angle 8$
- $\angle 1$ & $\angle 5$, $\angle 2$ & $\angle 6$, $\angle 3$ & $\angle 7$, and $\angle 4$ & $\angle 8$ are corresponding angles, therefore, $\angle 1 = \angle 5$, $\angle 2 = \angle 6$, $\angle 3 = \angle 7$, and $\angle 4 = \angle 8$
- $\angle 3$ & $\angle 6$ and $\angle 4$ & $\angle 5$ are interior angles on the same side of the transversal, therefore, $\angle 3 + \angle 6 = 180^{\circ}$, $\angle 4 + \angle 5 = 180^{\circ}$
- $\angle 3$ & $\angle 5$ and $\angle 4$ & $\angle 6$ are interior alternate angles, therefore, $\angle 3 = \angle 5$, $\angle 4 = \angle 6$
- $\angle 1$ & $\angle 7$ and $\angle 2$ & $\angle 8$ are alternate exterior angles, therefore, $\angle 1 + \angle 7 = 180^{\circ}$ and $\angle 2 + \angle 8 = 180^{\circ}$
How Do You Know If Lines Are Parallel?
To check whether the two given lines are parallel, you can use any of the following criteria.
- 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 interior angles on the same side of the transversal are supplementary.
Practice Problems
- What are parallel lines?
- How many angles are formed when a pair of parallel lines is intersected by a transversal?
- Name the pair of angles formed when a transversal intersects a pair of parallel lines.
- What are the different conditions that you can use to check whether a pair of lines are parallel or not?
FAQs
What are parallel lines in geometry?
Two or more lines are called parallel lines if they are always the same distance apart and never meet. The symbol used to denote parallel lines is ||. For example, AB||CD means line AB is parallel to line CD.
What are the types of angles formed in parallel lines and transversal?
When any two parallel lines are intersected by a transversal, the pair of angles formed are corresponding angles, alternate interior angles, alternate exterior angles, and interior angles on the same side of the transversal.
Are parallel lines equal in length?
No, parallel lines may not be equal in length but they should be the same distance apart.
What are the rules to verify that pair of lines are parallel?
To check whether the two given lines are parallel, you can use any of the following criteria.
a) Any two lines are said to be parallel if the corresponding angles so formed are equal.
b) Any two lines are said to be parallel if the alternate interior angles so formed are equal.
c) Any two lines are said to be parallel if the alternate exterior angles so formed are equal.
d) Any two lines are said to be parallel if the interior angles on the same side of the transversal are supplementary.
Conclusion
Parallel lines are lines that do not have a common intersection point and never cross paths with each other. When any two parallel lines are intersected by another line called a transversal, then pair of angles formed are corresponding angles, vertically opposite angles, alternate interior angles, alternate exterior angles, and interior angles on the same side of the transversal. In these pairs some are equal and some are supplementary and these properties are used to check whether a given pair of lines are parallel or not.
Recommended Reading
- What is Concurrent lines in Geometry – Definition, Conditions & Examples
- What is Half Line in Geometry – Definition, Properties & Examples
- What is a Perpendicular Line in Geometry – Definition, Properties & Examples
- Difference Between Axiom, Postulate and Theorem
- Lines in Geometry(Definition, Types & Examples)
- What Are 2D Shapes – Names, Definitions & Properties
- 3D Shapes – Definition, Properties & Types
