∑Richard Hummel Math

Angles

The measure of rotation between two rays sharing a common endpoint — the foundation of shape and direction.

Practice exercises →
Angle types are classified by their measure
45°
Acute
less than 90°
90°
Right
exactly 90°
130°
Obtuse
between 90° and 180°
180°
Straight
exactly 180°
Vertical
opposite angles match
Definition

An angle is formed when two rays share the same starting point, called the vertex. The rays are the sides of the angle. An angle measures the amount of rotation from one ray to the other.

Angles are measured in degrees. One full rotation is 360°360°.

Common angle types:

  • Acute angle: between 0°0° and 90°90°
  • Right angle: exactly 90°90° (the little square symbol marks these)
  • Obtuse angle: between 90°90° and 180°180°
  • Straight angle: exactly 180°180° — a flat line
Definition

Special angle pairs:

  • Complementary angles add up to 90°90°
  • Supplementary angles add up to 180°180°
  • Vertical angles are the opposite angles formed when two lines cross — they are always equal
Finding a missing angle

Angles AA and BB are supplementary. If A=65°A = 65°, find BB.

Since A+B=180°A + B = 180°: B=180°−65°=115°B = 180° - 65° = 115°

Try it

Two angles are complementary. One angle is three times the other. Find both angles.

Solution

Let the smaller angle be xx. Then the larger is 3x3x.

x+3x=90°x + 3x = 90°, so 4x=90°4x = 90°, giving x=22.5°x = 22.5° and 3x=67.5°3x = 67.5°.

Related concepts

Geometry· Foundations
Points, Lines & PlanesThe undefined building blocks of geometry — zero-dimensional locations, one-dimensional paths, and two-dimensional flat surfaces.
Geometry· Triangles
TrianglesThree-sided polygons with rich properties connecting angles, sides, area, and similarity.
Geometry· Trigonometry
Trigonometric RatiosSine, cosine, and tangent as ratios of sides in right triangles — the bridge between angles and lengths.

Related reading

Geometry· Foundations
Why Euclid Still MattersHow a 2,300-year-old list of five postulates gave us the template for all mathematical proof — and what happened when someone questioned the fifth one.
Geometry· Foundations
From Flat to CurvedWhat breaks when you do geometry on a sphere instead of a flat plane — and why that matters for GPS, general relativity, and the shape of the universe.