A corollary is a result widely used in geometry to indicate an immediate result of something already proven. Corollaries generally appear in geometry after a theorem has been proved.
Because they are a direct result of a proven theorem or a known definition, the corollaries do not require proof. These are very easy results to verify and therefore their proof is omitted.
Corollaries are terms that are mostly found in the realm of mathematics. But it is not limited to being used only in the area of geometry.
The word corollary comes from the Latin Corollarium, and is commonly used in mathematics, having a greater appearance in the areas of logic and geometry.
When an author uses a corollary, he is saying that this result can be discovered or deduced by the reader himself, using as a tool some previously explained theorem or definition.
Examples of Corollaries
Below are two theorems (which will not be proved), each followed by one or more corollaries that are deduced from said theorem. In addition, a short explanation of how the corollary is demonstrated is attached.
Theorem 1
In a right triangle, it is true that c² = a² + b², where a, b and c are the legs and the hypotenuse of the triangle respectively.
Corollary 1.1
The hypotenuse of a right triangle is longer than any of the legs.
Explanation: having c² = a² + b², it can be deduced that c²> a² and c²> b², from which it is concluded that «c» will always be greater than «a» and «b».
Theorem 2
The sum of the interior angles of a triangle is equal to 180º.
Corollary 2.1
In a right triangle, the sum of the angles adjacent to the hypotenuse is equal to 90º.
Explanation: in a right triangle there is a right angle, that is, its measure is equal to 90º. Using theorem 2 we have that 90º, plus the measures of the other two angles adjacent to the hypotenuse, is equal to 180º. By solving, it will be obtained that the sum of the measures of the adjacent angles is equal to 90º.
Corollary 2.2
In a right triangle the angles adjacent to the hypotenuse are acute.
Explanation: using corollary 2.1 we have that the sum of the measures of the angles adjacent to the hypotenuse is equal to 90º, therefore, the measure of both angles must be less than 90º and, therefore, these angles are acute.
Corollary 2.3
A triangle cannot have two right angles.
Explanation: if a triangle has two right angles, then adding the measures of the three angles will give a number greater than 180º, and this is not possible thanks to theorem 2.
Corollary 2.4
A triangle cannot have more than one obtuse angle.
Explanation: if a triangle has two obtuse angles, adding their measures will give a result greater than 180º, which contradicts Theorem 2.
Corollary 2.5
In an equilateral triangle the measure of each angle is 60º.
Explanation: an equilateral triangle is also equiangular, therefore, if "x" is the measure of each angle, then adding the measure of the three angles will obtain 3x = 180º, from which it is concluded that x = 60º.
References
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- Kinsey, L., & Moore, TE (2006). Symmetry, Shape and Space: An Introduction to Mathematics Through Geometry. Springer Science & Business Media.
- M., S. (1997). Trigonometry and Analytical Geometry. Pearson Education.
- Mitchell, C. (1999). Dazzling Math Line Designs. Scholastic Inc.
- R., MP (2005). I draw 6th. Progress.
- Ruiz, Á., & Barrantes, H. (2006). Geometries. Editorial Tecnologica de CR.
- Viloria, N., & Leal, J. (2005). Plane Analytical Geometry. Editorial Venezolana CA