Parallel Lines is the topic of International GCSE, O level Mathematics (Specification B) Geometry. We will be covering all math topics gradually. Please subscribe and turn on your bell button to get all updates.
Parallel Lines
Parallel lines are the lines that do not intersect or meet each other at any point in a plane. They are always parallel and are at equidistant from each other. Parallel lines are non-intersecting lines. We can also say Parallel lines meet at infinity.
Also, when a transversal intersects two parallel lines, then pairs of angles are formed, such as:
- Corresponding angles
- Alternate interior angles
- Alternate exterior angles
- Vertically opposite angles
- Linear pair
If two lines are intersecting each other at a point, in a plane, they are called intersecting lines. If they meet each other at 90 degrees, then they are called perpendicular lines.
Definition
Two lines are said to be parallel when they do not meet at any point in a plane. Lines which do not have a common intersection point and never cross path with each other are parallel to each other. The symbol for showing parallel lines is ‘||’.
Two lines which are parallel are represented as:
This means that line AB is parallel to CD.
The perpendicular distance between the two parallel lines is always constant.
In the figure shown above, the line segments PQ and RS represent two parallel lines as they have no common intersection point in the given plane. Infinite parallel lines can be drawn parallel to lines PQ and RS in the given plane.
Pairs of Angles
Lines can either be parallel or intersecting. When two lines meet at a point in a plane, they are known as intersecting lines. If a line intersects two or more lines at distinct points then it is known as a transversal line.
In figure 2, line l intersects lines a and b at points P and Q respectively. The line l is the transversal here.
∠1,∠2,∠7 and ∠8 are the exterior angles and ∠3,∠4,∠5 and ∠6 denote the interior angles.
The angle pairs formed due to intersection by a transversal are named as follows:
- Corresponding Angles: ∠1 and ∠6; ∠4 and ∠8; ∠2 and ∠5; ∠3 and ∠7 are the corresponding pair of angles.
- Alternate Interior Angles: ∠4 and ∠5 ; ∠3 and ∠6 denote the pair of alternate interior angles.
- Alternate Exterior Angles: ∠1 and ∠7; ∠2 and ∠8 are the alternate exterior angles.
- Same side Interior Angles: ∠3 and ∠5; ∠4 and ∠6 denote the interior angles on the same side of the transversal or co-interior or consecutive interior angles.
If the lines a and b are parallel to each other as shown, then the following axioms are given for angle pairs of these lines.
Properties of Parallel Lines
As we have already learned, if two lines are parallel, they do not intersect, on a common plane. Now if a transversal intersects two parallel lines, at two distinct points, then there are four angles formed at each point. Hence, below are the properties of parallel lines with respect to transversals.
- Corresponding angles are equal.
- Vertical angles/ 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.
