Points, Lines, and Angles

The Basic Objects of Geometry

In geometry, we build everything from a few basic ideas that are not defined in terms of simpler concepts. These are called geometric primitives. The most important primitives at this level are:

• Points
• Lines
• Angles

They are not “objects you can pick up,” but idealized concepts we use to describe shape, position, and size.

Points

A point represents an exact location in space, with no length, width, or thickness. It is purely a position.

In diagrams, we draw a point as a small dot and label it with a capital letter, such as $A$, $B$, or $P$.

• We refer to a point by its label: “point $A$”.
• Points are used to mark locations, ends of line segments, vertices (corners) of angles and polygons, and so on.

Important ideas involving points:

• Collinear points:
  Points that lie on the same straight line. For example, points $A$, $B$, and $C$ are collinear if there exists a line that passes through all three.
• Non-collinear points:
  Points that do not lie on a single straight line. Three non-collinear points can determine a triangle.

You can think of a point as “where” something is, not “how big” it is.

Lines and Related Concepts

A line extends forever in two opposite directions, has no thickness, and is completely straight.

In drawings, we show only a portion of a line with arrowheads on both ends to indicate that it continues without end.

Lines

We usually name a line in one of these ways:

• Using two distinct points on the line, such as line $AB$ (also written $\overleftrightarrow{AB}$).
• Using a single lowercase letter, like line $\ell$.

Key properties of a line:

• It is straight (no curves).
• It extends infinitely in both directions.
• Through any two distinct points, there is exactly one line.

When we say “point $P$ lies on line $\ell$”, we mean that $P$ is one of the (infinitely many) points that make up that line.

Line Segments

A line segment is part of a line with two endpoints and a finite length.

• If $A$ and $B$ are points, the segment with endpoints $A$ and $B$ is denoted $\overline{AB}$.
• Unlike a full line, a segment does not extend indefinitely; it stops at its endpoints.
• Segments can be measured; the length of segment $\overline{AB}$ is often written as $AB$ (without the bar) and is a number.

Segments are used to model fixed distances, sides of polygons, and edges of geometric figures.

Rays

A ray starts at one point and extends forever in one direction.

• If $A$ is the starting point and $B$ is another point along its direction, the ray is denoted $\overrightarrow{AB}$.
• The first letter is always the endpoint; the second letter shows the direction along the ray.
• A ray has:
• One endpoint.
• No length limit in its extending direction.

Rays are important when talking about angles, since an angle can be formed by two rays with a common endpoint.

Opposite Rays

Two rays are opposite rays if:

• They share the same endpoint, and
• They go in exactly opposite directions, forming a straight line.

For example, if $A$ is the common endpoint, and one ray goes through $B$ while the other goes through $C$, forming a straight line, then $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are opposite rays.

Opposite rays together form a straight line.

Intersecting, Parallel, and Perpendicular Lines

Lines are often compared based on how they relate to each other.

Intersecting Lines

Two lines are intersecting if they meet (cross) at exactly one point.

• The point where they meet is called the point of intersection.
• Intersecting lines form one or more angles at their intersection point.

Parallel Lines

Two lines in the same plane are parallel if they never meet, no matter how far they are extended.

• Notation: $ \ell \parallel m $ means line $\ell$ is parallel to line $m$.
• Parallel lines are everywhere the same distance apart.

Parallel line ideas are used heavily when working with polygons, especially triangles and quadrilaterals.

Perpendicular Lines

Two lines are perpendicular if they intersect to form a right angle (an angle of $90^\circ$).

• Notation: $ \ell \perp m $ means line $\ell$ is perpendicular to line $m$.
• At the intersection, the angles formed are all right angles.

Perpendicularity is central when describing squares, rectangles, and “height” or “altitude” in geometric figures.

Angles

An angle is formed by two rays that share a common endpoint.

• The common endpoint is called the vertex of the angle.
• The rays are called the sides (or arms) of the angle.

If the vertex is point $A$ and the rays go through points $B$ and $C$, the angle can be named:

• $\angle BAC$ (vertex in the middle),
• $\angle CAB$,
• or simply $\angle A$ if there is no ambiguity.

Measuring Angles

An angle measures how much one ray must be rotated around the vertex to line up with the other ray.

• The unit most commonly used is the degree, denoted by the symbol $^\circ$.
• A full turn around a point is $360^\circ$.
• A half turn (straight line) is $180^\circ$.
• A quarter turn is $90^\circ$.

Basic categories of angles by size:

• Acute angle: greater than $0^\circ$ and less than $90^\circ$.
• Right angle: exactly $90^\circ$.
• Obtuse angle: greater than $90^\circ$ and less than $180^\circ$.
• Straight angle: exactly $180^\circ$ (looks like a straight line).

These classifications help describe the shape and properties of geometric figures.

Angle Notation and Symbols

Typical ways to write or talk about angles include:

• $\angle ABC$ — angle with vertex at $B$, sides passing through $A$ and $C$.
• $m\angle ABC$ — the measure of angle $ABC$ (a number, like $45^\circ$).
• A small square at the vertex in a diagram often marks a right angle ($90^\circ$).

If angles are labeled with numbers or letters inside them (like $\angle 1$, $\angle x$), we may refer to them directly as angle $1$ or angle $x$.

Basic Relationships Between Angles

Once we have angles, we can compare and relate them. Some important relationships that show up frequently in geometry are:

Adjacent Angles

Two angles are adjacent if:

• They share a common vertex,
• They share a common side (ray),
• Their interiors do not overlap.

You can think of adjacent angles as “next to each other,” touching at the vertex and along one side.

Linear Pair

A linear pair is a special pair of adjacent angles whose non-common sides form a straight line.

• The two angles in a linear pair fit together to form a straight angle.
• Their measures add up to $180^\circ$.

Linear pairs are a key way that straight lines and angle measures interact.

Vertical Angles

When two lines intersect, they form four angles. Opposite angles in this intersection are called vertical angles.

• Vertical angles are not adjacent; they “face” each other.
• Each pair of vertical angles has the same measure.

Understanding vertical angles helps in solving many angle-measure problems involving intersecting lines.

Complementary and Supplementary Angles

Two angles relate in special ways if their measures add to important totals:

• Complementary angles: their measures add to $90^\circ$.
  If $\angle A$ and $\angle B$ are complementary, then
  $$ m\angle A + m\angle B = 90^\circ. $$
• Supplementary angles: their measures add to $180^\circ$.
  If $\angle C$ and $\angle D$ are supplementary, then
  $$ m\angle C + m\angle D = 180^\circ. $$

These relationships appear often when working with right angles and straight lines.

Bringing It All Together

Points, lines, and angles form the language in which most of geometry is written:

• Points locate positions.
• Lines (and their parts: segments and rays) describe straight paths and connections.
• Angles measure how lines and rays meet or turn.

Later geometric topics—such as triangles, polygons, circles, and more advanced theorems—are built on these basic objects and their relationships. Understanding the meaning and notation of points, lines, and angles is therefore an essential first step in studying geometry.