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.
| 1. | What are the Properties of Parallel Lines? |
| 2. | Solved Examples |
| 3. | Practice Questions |
| 4. | FAQs |
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:
- 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.
- 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.
- 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
Solved Examples on Properties of Parallel Lines
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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.
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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°
FAQs on Properties of Parallel Lines
What are the Properties of Parallel Lines?
The properties of parallel lines are - transitive property, symmetry property and angles property.
What are the Properties of Parallel Lines Cut by a Transversal?
Properties of parallel lines cut a transversal are as follows:
- The pairs of corresponding angles are equal.
- The pairs of vertically opposite angles are equal.
- The pairs of alternate interior angles are equal.
- The pairs of alternate exterior angles are equal.
- The pair of interior angles formed on the same side of the transversal holds the supplementary angle property.
What is Special about Parallel Lines?
The special characteristic of parallel lines is that they never meet. They are always equidistant, even when extended till infinity.
What are the Angle Properties of Parallel Lines?
Angle properties of parallel lines correspond to the pairs of angles: pairs of corresponding angles, pairs of alternate interior angles, and pairs of alternate exterior angles, which are so formed when a transversal cuts two parallel lines.
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Corresponding angle pairs are equal, making an F shape.
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Alternate angle pairs are equal, making a Z shape (which can also be back to front).
Which Properties do Best Describe the Coordinate Graph of Two Parallel Lines?
The property which best describes the coordinate graph of two parallel lines is that two distinct parallel lines have the same slope but they have different y-intercepts.
What Properties are Specific to Parallel Lines and Perpendicular Lines?
The two lines, either both horizontal or both vertical, are in the same plane and have the same slope, then they are considered parallel. These lines never intersect. On the other hand, if the two lines in the same plane intersect at a right angle, then they are considered perpendicular.
How is the Transitive Property of Parallel Lines Similar to the Transitive Property of Congruence?
The transitive property of parallel lines is similar to the transitive property of congruence in the way that they both satisfy the condition - if A=B and B=C then A=C.