Angles – Measuring Turns and Corners
Angles are everywhere — in the corners of rooms, the hands of a clock, and the slope of a ramp. Understanding angles is one of the most practical skills in geometry.
We owe the 360-degree circle to the ancient Babylonians, who used a base-60 (sexagesimal) number system roughly 4,000 years ago; 360 divides evenly by so many small numbers (2, 3, 4, 5, 6, 8, 9, 10, 12...) that it made calculating fractions of a turn far easier than most other choices, and some historians link the figure to their approximation of 360 days in a year. The protractor, the simple tool still used in classrooms today to measure angles in degrees, was not developed in its familiar semicircular form until the 16th and 17th centuries, originally for navigation and surveying. Angles remain essential far beyond the classroom: architects calculate the angle of a roof pitch to shed rain and snow safely, and a basketball player instinctively judges the release angle of a shot to arc the ball cleanly through the hoop.
What Is an Angle?
An angle is the amount of turn between two rays that share the same endpoint, called the vertex. Angles are measured in degrees (°). One full turn is 360°.
Types of Angles
| Type | Size | Description |
|---|---|---|
| Zero angle | 0° | No turn at all |
| Acute angle | Between 0° and 90° | Smaller than a right angle |
| Right angle | Exactly 90° | A perfect corner, shown with a small square |
| Obtuse angle | Between 90° and 180° | Larger than a right angle but not flat |
| Straight angle | Exactly 180° | A flat line |
| Reflex angle | Between 180° and 360° | More than a half turn |
| Full angle | 360° | One complete rotation |
Angle Relationships
| Relationship | Rule | Example |
|---|---|---|
| Complementary angles | Add up to 90° | 35° and 55° |
| Supplementary angles | Add up to 180° | 110° and 70° |
| Angles on a straight line | Add up to 180° | Angles formed when a line is split by a ray |
| Angles at a point | Add up to 360° | All angles meeting at one vertex |
| Vertically opposite angles | Equal to each other | Formed when two lines cross |
Angles and Parallel Lines
When a straight line (called a transversal) crosses two parallel lines, it creates several pairs of equal or supplementary angles.
| Angle Pair | Property |
|---|---|
| Alternate interior angles | Equal (Z-angles) |
| Co-interior (same-side) angles | Add up to 180° (C-angles) |
| Corresponding angles | Equal (F-angles) |
| Vertically opposite | Equal |
Worked Examples
90 − 37 = 53°.
x + 3x = 180. 4x = 180. x = 45°. The angles are 45° and 135°.
Alternate interior angles are equal. Answer: 65°.
Key Takeaways
- Angles are measured in degrees; a full turn is 360°.
- Acute < 90°; Right = 90°; Obtuse is between 90° and 180°; Reflex > 180°.
- Complementary angles sum to 90°; supplementary angles sum to 180°.
- Vertically opposite angles are equal; alternate angles in parallel lines are equal.