Properties of Parallel Lines

Parallel lines are the set of lines that lie on the same plane but never intersect each other, even if you extend them infinitely. The set of parallel lines is denoted by the symbol ||. Any two parallel lines are always equidistant. Let's explore the properties of parallel lines.

What are the Properties of Parallel Lines?

Two lines in a plane are said to be parallel if they do not intersect when extended infinitely in both directions. Two straight lines are said to be parallel if their slopes are equal and they have different y-intercepts. Given below are a few important properties of parallel lines that actually reflect the characteristics of parallel lines.

Transitive Property of Parallel Lines

The lines which are parallel to the same line are also parallel to each other. This property is referred to as the transitive property of parallel lines. It also holds good for more than 2 lines, such as if the line k is parallel to line l and line l is parallel to line m then line k is parallel to the line m.

Symmetric Property of Parallel Lines

The symmetric property of parallel lines states the two or more parallel lines are symmetric. If line1 is parallel to line2, then line2 is also parallel to line1. In the image above, if k|| l, then l || k. Note: As per Euclid's tenets, parallelism is not a reflexive relation and thus in a way fails to be an equivalence relation.

Properties of Parallel Lines Cut by a Transversal

For the parallel lines cut by a transversal, the following properties hold true:

1. Two lines cut by a transversal line are parallel if the corresponding angles so formed are equal. In general, the corresponding angles are in relative positions and lie along the same side of the transversal.
2. Two lines cut by a transversal line are parallel if the alternate interior angles so formed are equal. In general, the pairs of the alternate interior are found in the inner side but lie on the opposite sides of the transversal.
3. Two lines cut by a transversal line are parallel if the alternate exterior angles so formed are equal. In general, the pairs of alternate exterior angles are found on the outer side but lie opposite each other.

Important Notes

When a transversal intersects two parallel lines:

• The corresponding angles are equal.
• The vertically opposite angles are equal.
• The alternate interior angles are equal.
• The alternate exterior angles are equal.
• The pair of interior angles on the same side of the transversal is supplementary.

Related Topics

• Transversal and Related Angles
• Linear Pairs of Angles

Solved Examples on Properties of Parallel Lines

Example 1: Determine if the lines p, q, and r are parallel. Solution: Here, the pair of corresponding angles are equal, that is 65° and the pair of alternate exterior angles are equal, that is 115°. Therefore, with the angles property of parallel lines, we can conclude that the lines p, q, and r are parallel.

Example 2: l and m are two parallel lines, P is the transversal. Find the measures of angle A and angle B. Solution: It is given that the lines, l and m are parallel intersected by the transversal P. Thus, we can say that a = b (alternate interior angles). Also, a + 140° = 180° (pair of supplementary angles). This implies, a = (180-140)° = 40°. Therefore, a and b = 40°

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