We use mental images of parallel lines, squares and circles as our definitions. In common language, we begin with an object or idea. A definition is no more then an attempt to describe in words the preexisting object or idea. For example, a Dog is something that exists in the real world. In geometry, the definition is primary.
Geometry begins with definitions of abstract, unvisualized objects.
For example, parallel lines DO NOT exist in the world. This distinction is very difficult to understand and is the source of much confusion about geometric proofs. A study of Hyperbolic Geometry helps us to break away from our pictorial definitions by offering us a world in which the pictures are all changed - yet the exact meaning of the words used in each definition remain unchanged. Hyperbolic Geometry helps us focus on the importance of words. Several interactive software tools have been developed which allow students to explore Euclidean geometry.