Parallel and perpendicular lines

Understanding Parallel and Perpendicular Lines

In this chapter we focus on two special relationships between lines: being parallel and being perpendicular. We will describe them both geometrically (by their angles and how they meet) and algebraically (using equations and slopes), but without going into topics that belong to coordinate geometry or later algebra chapters in detail.

Parallel Lines

Two lines in a plane are called parallel if they never meet, no matter how far they are extended in either direction.

Informally: they are always the same distance apart.

Basic properties of parallel lines

Suppose we have two distinct lines $\ell_1$ and $\ell_2$ in a plane.

They are parallel if:

• They do not intersect (do not have any point in common),
• Or they are exactly the same line (sometimes we say “coincident”; in many school contexts, “parallel” is reserved for distinct non-intersecting lines).

We usually write:

• $\ell_1 \parallel \ell_2$ to mean “$\ell_1$ is parallel to $\ell_2$.”

Key ideas:

• If two lines are parallel, every point on one line is at the same distance from the other line.
• Parallelism is a property within a plane: any line parallel to a given line lies in the same plane as that line (or in a parallel plane in 3D, but that is beyond our scope here).

Parallel line segments and rays

• Two segments are parallel if they lie on parallel lines.
• Two rays are parallel if they lie on parallel lines and point in “similar directions” (they do not cross).

For many practical purposes (drawing, measuring, building), we care about parallel segments: for example, opposite sides of a rectangle are parallel segments.

Notation in diagrams

In diagrams, parallel lines are often marked with matching arrow symbols:

• One pair of arrows on one pair of opposite sides,
• A different style (double arrows) on another pair, if there is more than one pair of parallel sides.

Example: In a rectangle $ABCD$,

• $AB \parallel CD$,
• $BC \parallel AD$.

Perpendicular Lines

Two lines in a plane are called perpendicular if they meet at a right angle.

A right angle is an angle that measures $90^\circ$.

Basic properties of perpendicular lines

Suppose we have two lines $m$ and $n$.

They are perpendicular if:

• They intersect,
• And they form four right angles at their intersection.

We usually write:

• $m \perp n$ to mean “$m$ is perpendicular to $n$.”

Key ideas:

• When two lines are perpendicular, the angles they form at the intersection are all congruent (all $90^\circ$).
• A perpendicular pair gives a clear sense of “vertical” and “horizontal” directions, commonly used in grids, floors and walls, and graph paper.

Perpendicular segments and rays

• Two segments are perpendicular if the lines containing them are perpendicular.
• Two rays are perpendicular if their sides lie on perpendicular lines and they share a common endpoint (the vertex of the right angle).

In diagrams, right angles are usually marked with a small square at the vertex.

Example: If segment $AB$ is perpendicular to segment $CD$, we may write:

• $\overline{AB} \perp \overline{CD}$,
  and show a little square at their intersection.

Parallel and Perpendicular Lines with Angles

The parent chapter introduced angle types. Here we focus specifically on how angles relate to parallel and perpendicular lines.

Angles formed by perpendicular lines

If line $m$ is perpendicular to line $n$ at point $P$, then:

• The four angles around $P$ have measures:
  $^\circ, 90^\circ, 90^\circ, 90^\circ.$$
• Each pair of adjacent angles are complementary (their measures add up to $180^\circ$), but here each individual angle is exactly $90^\circ$.
• Opposite angles at the intersection are vertical angles and are equal. In this special case, all vertical angles are right angles.

This structure—four equal right angles at a point—is the defining picture of perpendicularity.

Angles and parallel lines (conceptual overview)

In a more advanced chapter on angles with parallel lines, you will study special angle pairs (corresponding, alternate interior, etc.) created when a third line cuts across two parallel lines. Here we only state the key idea:

• When a single line (called a transversal) crosses two parallel lines, many pairs of angles turn out to be equal, or add to $180^\circ$.
• These relationships are characteristic of parallel lines and are often used to prove that lines are parallel.

We do not go into full detail here, because the specific names and theorems for these angle pairs belong to another chapter.

Parallel and Perpendicular Lines in Coordinate Form (Ideas Only)

A later course on analytic geometry and algebra will study lines using equations. There, slope will be the main tool for describing parallel and perpendicular lines.

Here is the essential connection, without proofs or technical detail:

• Lines can be represented by equations such as
  $$y = mx + b,$$
  where $m$ is the slope and $b$ is a constant.
• The slope $m$ describes the steepness and direction of the line.

Using slope:

• Two non-vertical lines are parallel if they have the same slope.
• Two non-vertical lines are perpendicular if the product of their slopes is $-1$ (this condition will be fully explored in algebra/geometry chapters on slope).

Vertical and horizontal lines fit nicely into this picture:

• A horizontal line has slope $0$.
• A vertical line has no (finite) slope and is perpendicular to every horizontal line.

We mention this connection only to link the geometric idea (shape and angle) with the algebraic idea (slope); the detailed work with equations is reserved for later chapters.

Constructing and Identifying Parallel and Perpendicular Lines

You will often need to:

• Draw a line parallel to a given line through a point.
• Draw a line perpendicular to a given line through a point.
• Recognize whether lines appear parallel or perpendicular in a figure.

Visual and practical recognition

In diagrams or real-world objects:

• Parallel lines:
• Look like “tracks” that never get closer or farther apart.
• Examples: the sides of a straight railway track, edges of a perfectly rectangular book, stair railings.
• Perpendicular lines:
• Look like a perfect “L” or a “plus” sign at their intersection.
• Examples: the corner edges of a book, floor meeting a wall (ideally), crosshairs on a scope.

In more precise geometry:

• A right angle marker (small square) is used to show perpendicularity.
• Matching arrow markers are used to show parallelism.

Using tools (conceptual)

Using tools such as a ruler, a right-angle set square, or a compass:

• To draw a line perpendicular to a given line through a point:
• Place the right-angle tool so one side lies along the given line and the corner passes through the point.
• Draw along the other side of the right angle.
• To draw a line parallel to a given line through a point:
• Use a tool that keeps the same angle or distance (e.g. a set square sliding along a ruler) so that the new line never gets closer to or farther from the original.

Again, the step-by-step use of instruments belongs to practical geometry; here we only emphasize that these constructions are built around the ideas of equal distance (parallel) and right angle (perpendicular).

Relationships Between Parallel and Perpendicular Lines

Two important combined ideas often appear together:

1. Perpendicular to the same line
   If two lines are both perpendicular to the same line (and all three lie in the same plane), then the two lines are parallel to each other.

Intuitively:

• Imagine a floor and two vertical poles.
• If both poles are standing perfectly upright (perpendicular to the floor), then the poles are parallel to each other.

2. Parallel and perpendicular combination in shapes
   Many familiar shapes are defined using both concepts:
• A rectangle has opposite sides parallel and adjacent sides perpendicular.
• A square has all sides equal in length, opposite sides parallel, and adjacent sides perpendicular.

Recognizing these patterns helps you identify and classify shapes quickly based on their side and angle relationships.

Summary

• Parallel lines in a plane never intersect and stay the same distance apart. We write $\ell_1 \parallel \ell_2$.
• Perpendicular lines intersect to form four right angles ($90^\circ$ each). We write $m \perp n$.
• Parallel and perpendicular relationships can apply to whole lines, segments, and rays, based on the lines that contain them.
• Right angle markers ($\square$ at the vertex) indicate perpendicularity; matching arrow marks indicate parallelism in diagrams.
• In later chapters using coordinates and algebra, parallelism and perpendicularity will be described using equations and slopes, connecting these geometric ideas with algebraic ones.