What theory studies the properties and relationships of geometric figures in two and three dimensions?

The theory that studies the properties and relationships of geometric figures in two and three dimensions is Euclidean Geometry. This branch of mathematics is based on the work of the ancient Greek mathematician Euclid, who formulated a system of geometry that includes definitions, postulates, and proofs that describe the relationships between points, lines, angles, and shapes in a flat (two-dimensional) and three-dimensional space.

Euclidean Geometry provides a foundation for understanding geometric concepts such as congruence, similarity, area, volume, and the properties of various shapes such as triangles, circles, and polygons. It relies on a set of axioms that are universally accepted, allowing mathematicians to derive further truths about geometrical structures.

Other options like Non-Euclidean Geometry explore geometries that deviate from Euclid's axioms, primarily in curved spaces. Calculus focuses on the study of change and motion rather than static geometric properties, while Statistics involves the analysis of data rather than geometric relationships. Thus, Euclidean Geometry is the appropriate choice when discussing the properties and relationships of geometric figures in two and three dimensions.