Parallel and Perpendicular Lines

Parallel lines are two straight lines that will never meet. They are on the same plane and will never intersect.

They are commonly marked with a symbol that looks like this: $||$

Try to imagine a straight path leading from infinity to infinity. The edges of the path will never meet, like parallel lines.

Meet the properties of parallel lines:

• The distance between all points on one parallel line to the second parallel line is identical
• The slopes of parallel lines are identical - meaning their steepness - how much they are inclined up and down is the same in both - this is what causes them to never meet or intersect.

If we look closely, we can identify that parallel lines form various different shapes such as parallelogram, rhombus, square, and rectangle.
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Perpendicular lines are lines that are at right angles to each other and form a right angle of $90$ degrees between them.
Upon closer inspection, we can identify that perpendicular lines make up a wide variety of geometric shapes such as squares, rectangles, and right triangles.

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Angles in parallel lines are angles created when a third line intersects the two given parallel lines.
To classify the angles created by the intersection, we need to identify if they are:

• Above the line - the pink part
• Below the line - the light blue part
• Right of the line - the red part
• Left of the line - the green part

Try to think of it as 2 buildings adjacent to each other. In these buildings, angles reside.
To classify them, you need to understand the side and floor of each angle.

• Located on the same side of the transversal line
• Located on the same "level" relative to the line
The property - corresponding angles between parallel lines are equal.

Here are corresponding angles for example:

The two marked angles are located on the left side and on the bottom floor - therefore they are corresponding and equal.
- Both are on the same side and on the same floor.

• Share a common vertex - located on the same vertex
• Located opposite to each other
The property - vertical angles between parallel lines are equal.

Here are examples of vertical angles:

The two marked angles are located on the same vertex and are opposite to each other, therefore they are vertical angles.
They share the same vertex.

• Adjacent to each other
• Located on the same line
The property - The sum of adjacent angles between parallel lines equals 180°.

Here are adjacent angles for example:

The two marked angles are on the same line - orange and are adjacent to each other, therefore they are adjacent angles.

• Not located on the same side of the intersecting line
• Not located on the same "level" in relation to the line
The property - alternate angles between parallel lines are equal.

Here are alternate angles for example:

The two marked angles are not on the same floor and not on the same side, therefore they are alternate angles.
The first angle in building A is on the right side on the upper floor, while the second angle in building B is on the left side on the lower floor.

• Located on the same side of the transversal line
• But not on the same "level" in relation to the line
The property - The sum of corresponding angles between parallel lines equals 180°.

Here are examples of one-sided angles:

The two marked angles are not on the same floor but are on the same side, therefore they are corresponding angles.
The first angle in building A is on the right side on the upper floor, while the second angle in building B is on the right side on the lower floor.

Note - The red angles in the figure above are exterior alternate angles because they are located on the outer side of the parallel lines.
Interior alternate angles are located on the inner side of the parallel lines:

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