Parallel Lines
What are Parallel Lines?
Parallel lines in geometry are two lines in the same plane that never intersect, no matter how far they are extended. They remain equidistant from each other at all points.
Properties of Parallel Lines
- Never Intersect: Parallel lines never cross or meet, no matter how far they are extended. They stay the same distance apart everywhere. For example, railroad tracks are parallel because they always stay evenly spaced.
- Equal Slopes: If you compare the slopes of two parallel lines, they will always be the same. This is true for lines on a graph, where slope measures how steep the line is.
- Equal Spacing: Parallel lines are always the same distance apart. This distance never changes, even if the lines are extended forever.
Angles Formed by Transversals
When a transversal intersects two parallel lines, it creates several types of angles. A transversal is just a line that crosses two or more other lines. The angles formed by this intersection have special relationships, which makes solving geometry problems easier.
- Corresponding Angles:
These are angles in the same position on each parallel line. For example, if one angle is above the parallel line on the left side of the transversal, its corresponding angle will also be above the other parallel line on the left side of the transversal.
Property: Corresponding angles are always equal. - Alternate Interior Angles:
These angles are on opposite sides of the transversal and are inside the parallel lines. For instance, if one angle is on the left side of the transversal inside the parallel lines, its alternate interior angle will be on the right side of the transversal inside the lines.
Property: Alternate interior angles are always equal. - Alternate Exterior Angles:
These are on opposite sides of the transversal but outside the parallel lines. For example, if one angle is on the left side of the transversal and above the parallel line, its alternate exterior angle will be on the right side of the transversal and above the other parallel line.
Property: Alternate exterior angles are always equal. - Consecutive Interior Angles (Also Called Same-Side Interior Angles):
These angles are on the same side of the transversal and inside the parallel lines. For example, both angles might be on the right side of the transversal between the parallel lines.
Property: Consecutive interior angles add up to
.