Corollary — Definition, Meaning & Examples
Corollary
A special case of a more general theorem which is worth noting separately. For example, the Pythagorean theorem is a corollary of the law of cosines.
See also
Example
Problem: Show that the Pythagorean theorem is a corollary of the law of cosines.
State the general theorem: The law of cosines states that for any triangle with sides a, b, c and angle C opposite side c:
c2=a2+b2−2abcosC
Apply the special case: Now consider a right triangle where angle C = 90°. Since cos 90° = 0, the term −2ab cos C vanishes:
c2=a2+b2−2abcos90°=a2+b2−2ab(0)
Simplify to get the corollary: The equation reduces immediately to the Pythagorean theorem:
c2=a2+b2
Answer: The Pythagorean theorem follows as a corollary of the law of cosines by setting the angle to 90°. No new proof technique was needed — just substitution into the already-proven general result.
Another Example
Problem: A theorem states: "If a quadrilateral is a rectangle, then its diagonals are equal in length." State a corollary for the special case of a square.
Identify the general theorem: The theorem applies to all rectangles: their diagonals are equal in length.
Recognize the special case: A square is a special type of rectangle (all angles are 90° and all sides are equal). Because a square is a rectangle, the theorem automatically applies to it.
State the corollary: Corollary: The diagonals of a square are equal in length. This needs no separate proof — it inherits the result directly from the rectangle theorem.
Answer: The diagonals of a square are equal in length. This is a corollary because a square is a special case of a rectangle, and the general theorem already covers rectangles.
Frequently Asked Questions
What is the difference between a corollary and a theorem?
A theorem is a significant mathematical statement that requires its own proof. A corollary is a result that follows almost immediately from a theorem, needing little or no extra work to prove. You can think of a corollary as a 'bonus' result you get for free once the main theorem is established.
Does a corollary need a proof?
Technically yes, but the proof is usually very short — often just a sentence or two explaining how the corollary follows from the parent theorem. In many textbooks, the proof of a corollary is as simple as substituting a specific value or noting that the situation is a special case of the general result.
Corollary vs. Lemma
A corollary comes *after* a theorem and follows from it as an easy consequence. A lemma comes *before* a theorem and serves as a stepping stone to help prove it. Both are proven statements, but they play opposite roles relative to the main theorem: a lemma builds up to the theorem, while a corollary flows down from it.
Why It Matters
Corollaries help organize mathematical knowledge efficiently. Instead of proving every useful result from scratch, mathematicians prove one powerful general theorem and then list its important special cases as corollaries. When you encounter a corollary in a textbook, it signals that the result is closely tied to the preceding theorem and that understanding the theorem unlocks many related facts at once.
Common Mistakes
Mistake: Thinking a corollary is just a restatement of the theorem in different words.
Correction: A corollary is a *different* statement that follows from the theorem, usually by restricting to a special case or drawing a specific consequence. It says something new, even if the proof is short.
Mistake: Believing a corollary is less true or less important than a theorem.
Correction: A corollary is just as rigorously true as the theorem it comes from. The label 'corollary' reflects how it was derived (easily, from an existing result), not its importance. The Pythagorean theorem is arguably more famous than the law of cosines, yet it can be viewed as a corollary of it.
Related Terms
- Theorem — The main result a corollary follows from
- Lemma — A helper result proven before a theorem
- Axiom — A starting assumption accepted without proof
- Postulate — Another term for an axiom in geometry
- Pythagorean Theorem — Classic corollary of the law of cosines
- Law of Cosines — General theorem yielding the Pythagorean corollary