Definition of a Corollary
A corollary in mathematics is a proposition, theorem, or statement that logically follows directly and easily from a previously proven theorem or proposition, often requiring little or no additional proof. It is essentially a direct consequence or an obvious deduction, making complex ideas more accessible.
Relationship to Theorems
Corollaries are derived from theorems that have already been established as true. They are not independent theorems but rather extensions or specific cases that become evident once the main theorem is understood and proven. They serve to highlight immediate implications.
Example in Geometry
A classic example is from geometry. Once the 'Angle Sum Theorem' (that the sum of angles in a triangle is 180 degrees) is proven, a corollary can be stated: 'Each angle in an equilateral triangle is 60 degrees.' This follows directly because an equilateral triangle has three equal angles, and 180 divided by 3 equals 60.
Importance in Mathematical Reasoning
Corollaries simplify mathematical reasoning by allowing deductions to be made quickly from established truths without having to re-prove every related statement. They highlight the immediate implications and applications of a major theorem, showing direct logical connections within mathematical structures.