Two straight lines, in the same plane, are said to be parallel if they do not intersect, no matter on how long they are extended on either sides.
A straight line that intersects two or more lines in a plane is called a transversal.
When two lines m and n are intersected by line l, then there are eight angles that are formed as shown in the figure. These angles can be grouped into different groups:
- Pairs of Corresponding Angles:
- Pairs of Alternate Interior Angles:
- Pair of Alternate Exterior Angles:
- Pair of Consecutive Interior Angles:
The opposite angles are equal. Therefore . These are vertically opposite angles (please refer to Lecture Notes Part 2).
Properties of Angles Associated with Parallel Lines
Now let’s extend this understanding to a pair of parallel lines that is intersected by another lines. Same logic, just that m and n are parallel lines.
- Corresponding Angles are equal:
Alternate Interior Angles are Equal:
- Alternate Exterior Angles are Equal:
- Consecutive Interior Angles are supplementary (i.e. the sum of the angles is 180°:
The converse of the above is also true:
- If the Corresponding Angles are equal, then the lines are parallel.
- If the alternate angles are equal, then the lines are parallel.
- If the consecutive interior angles are supplementary, then the lines are parallel.
- To intersecting lines cannot be both parallel to the same straight lines.
- Also, straight lines parallel to the same lines are also parallel to each other.
- Through a given point, there can only be one straight line parallel to a given line.