Angles in Geometry

Angles are all around us: in the way a door swings open, the shape of a roof, or even when you fold a sheet of paper. Understanding angles helps us make sense of the world we live in.

So, what exactly is an angle?

An angle is created when two rays share the same starting point, known as the vertex.

The rays, referred to as the sides of the angle, extend infinitely from this common point.

An angle represents one of the regions into which the plane is divided by these rays.

With the exception of the special case of a straight angle, one of the two angles is concave, while the other is convex.

An angle is classified as concave if its measure exceeds 180° and its interior includes the extended sides. Conversely, an angle is convex if its measure is 180° or less and does not include the extended sides.

In simple terms, an angle measures the amount of rotation needed to bring one side into alignment with the other.

Angles are typically measured in a counterclockwise direction.

To put it simply, it tells you how much you need to rotate one side of the angle to overlap it with the other.

Imagine opening a book on a table. The point where the two pages meet acts as the vertex of an angle, and the edges of the pages are its sides. The angle represents the space between these "pages." To close the book, you bring one side over to meet the other.

Ways to Represent an Angle

There are various ways to denote an angle.

One common method is using \( AÔB \), where \( A \) and \( B \) are points on the sides of the angle.

You can also use the symbol \( \angle AÔB \).

Another way to represent an angle is by referring to the rays \( a \) and \( b \) that form its sides, using the notation \( aÔb \).

A shorter notation is \( ab \), which highlights only the two sides of the angle, without specifying the vertex.

You can also represent the angle as \( Ĉ \), indicating just the vertex without mentioning the sides.

For simplicity, angles are often labeled with lowercase Greek letters such as \( \alpha \), \( \beta \), \( \gamma \), and so on.

These are the main ways to represent an angle.

How do we measure angles?

Angles are measured using tools like a protractor, a semicircular or circular device marked with degrees. It’s an essential tool in technical drawing and geometry.

Angles can be measured in different units: degrees, radians, or revolutions.

• Degrees
  A degree represents 1/360 of a complete turn of a circle. A full circle equals 360°, while half a circle measures 180°, and so on. This system dates back to the ancient Babylonians, who divided a circle into 360 equal parts.

  For example, think of a clock face divided into 360 segments instead of 60 minutes.

• Radians
  A radian is the angle formed when the arc length of a circle equals its radius.
• Revolutions
  A revolution represents a full 360° turn around a circle, equivalent to 2π radians.

Once you understand how angles are measured, you can explore their various types.

Types of Angles

Angles are categorized based on their size:

• Acute Angles
  These are small angles measuring less than 90 degrees, like the angle in a slice of pizza.
  
• Right Angles
  A right angle measures exactly 90 degrees, like the corner of a sheet of paper. Lines that form a right angle are called perpendicular.
  
• Obtuse Angles
  These angles are larger than a right angle but smaller than a straight angle, like the angle formed when your arms are opened but not fully extended.
  
• Straight Angles
  A straight angle measures 180 degrees. Its sides are opposite each other, forming a straight line. This is the angle created when you extend your arms in opposite directions.
  
• Full Angles
  A full angle measures 360°, completing a full rotation of a circle. For example, spinning around completely results in a 360-degree rotation, which is a full angle. In this case, the two angles overlap completely.
  
• Zero Angle
  A zero angle is formed when both sides of the angle overlap entirely. Unlike a full angle, it contains no points other than those lying directly on its sides.
  

Relationships Between Angles

When two angles share the same vertex, they can have specific relationships:

• Adjacent Angles
  Adjacent angles share one side, while their other sides extend in opposite directions from the same point.
  
• Linear Pair of Angles
  These angles are next to each other, sharing a vertex and a side, and their sum is always 180 degrees, forming a straight line. They are a special type of adjacent angles that add up to a straight angle.
  
• Complementary Angles
  Complementary angles are pairs of angles that add up to 90 degrees. No matter their individual sizes, together they form a perfect right angle. They don’t always have a shared side or a common vertex.
  
• Supplementary Angles
  These angles don’t need to be adjacent, but their sum is always 180 degrees.
  
• Explementary Angles
  Two angles are considered explementary if their sum is equal to a full circle, or 360 degrees.
  
• Opposite Angles
  Opposite angles are formed when two straight lines intersect, creating two pairs of equal angles. These angles are congruent and symmetrical. The extensions of an angle's sides create the sides of a different angle.
  

Angles aren’t just a mathematical concept—they’re a key part of how we perceive and interact with the world. They’re everywhere, shaping the things we do and the spaces we inhabit.

Angles have countless practical applications. In architecture and engineering, understanding angles is essential for designing and constructing buildings and structures. In astronomy, they’re crucial for calculating the positions and trajectories of celestial bodies, among many other uses.