Which theory explores the relationship between angles and the sides of triangles?

Trigonometry is the branch of mathematics that focuses specifically on the relationships between the angles and sides of triangles. This area of study is essential for understanding various properties of triangles, including the sine, cosine, and tangent functions, which relate the angles of a triangle to the ratios of its sides.

In trigonometry, for any given triangle, the relationships established allow for the calculation of unknown angles or sides when some measures are known. This principle is foundational not only in mathematics but also in fields such as physics, engineering, and navigation, where understanding the measures within triangles can lead to solutions in real-world applications.

The other theories mentioned, such as vector theory, function theory, and probability theory, do not focus on the specific relationship between angles and sides of triangles in the way that trigonometry does, making it the correct choice for exploring that relationship.