Angles and Triangles: Types, Rules & Formulas

Any time two lines meet, they form an angle - and once you connect three of those angles with three straight sides, you get a triangle, the simplest closed shape in geometry. The rules that govern angles and triangles aren't just classroom material, either. They're the backbone of trigonometry and show up throughout applied math, from architecture to navigation to computer graphics.

Types of Angles

Angles get grouped by how large they are, ranging from a sliver under 90° up to a full 360° turn:

Type	Measure	Example
Acute	Less than 90°	45°, 60°
Right	Exactly 90°	Corner of a square
Obtuse	Between 90° and 180°	120°, 150°
Straight	Exactly 180°	A flat line
Reflex	Between 180° and 360°	270°
Full rotation	Exactly 360°	Full circle

Key Angle Relationships

Complementary Angles

Two angles are complementary when their measures add up to 90°. Each one is called the complement of the other.

30° and 60° are complementary (30 + 60 = 90)

Supplementary Angles

Supplementary angles add up to 180°, which means together they form a straight line.

110° and 70° are supplementary (110 + 70 = 180)

Vertically Opposite Angles

Whenever two straight lines cross, the pair of angles sitting directly across from each other at that intersection are always equal.

Angles on a Straight Line

Any set of angles arranged along one straight line will always total 180°.

Angles at a Point

Angles arranged all the way around a single point add up to 360°, a full rotation.

Parallel Lines and Transversals

Cross two parallel lines with a third line (called a transversal) and a predictable set of angle relationships appears at each intersection:

Angle pair	Relationship
Corresponding angles	Equal (same position at each intersection)
Alternate interior angles	Equal (Z-shape between the lines)
Co-interior angles	Add up to 180° (C-shape / same side)

Angle Rules Specific to Triangles

Angle Sum of a Triangle

No matter the triangle's shape, its three interior angles always sum to exactly 180°:

A + B + C = 180°

If A = 50° and B = 70°, then C = 180 − 50 − 70 = 60°

Exterior Angle Theorem

An exterior angle, formed by extending one side of the triangle, always equals the sum of the two interior angles that aren't next to it:

Exterior angle = sum of the two opposite interior angles

Interior angles 40° and 65°:
Exterior angle = 40 + 65 = 105°

The Sine Rule

For a triangle with sides a, b, and c sitting opposite angles A, B, and C respectively:

a/sin(A) = b/sin(B) = c/sin(C)

Find side b: a = 8, A = 30°, B = 45°
b = 8 × sin(45°) / sin(30°) = 8 × 0.707 / 0.5 ≈ 11.3

The Cosine Rule

This one comes in handy when you know two sides and the angle between them, or when you know all three sides and need to find an angle:

c² = a² + b² − 2ab·cos(C)

a = 5, b = 7, C = 60°:
c² = 25 + 49 − 2(5)(7)(0.5) = 74 − 35 = 39
c = √39 ≈ 6.24

Angle Bisector

An angle bisector splits an angle into two equal halves. Every triangle contains three of them, one from each vertex, and all three always cross at a single shared point known as the incenter.

Solving Triangles Automatically

SolverCalc's triangle calculator will solve any triangle once you feed it a workable combination of sides and angles. It applies the sine and cosine rules behind the scenes and sketches a live diagram of the result, so you can see the triangle take shape rather than just reading off numbers.

Understanding Angles and Triangles: Types and Rules