A theorem is a math statement that has been proved true using definitions, postulates and statements already proved. Once proved, it is true in every case it covers, so you can use it to solve problems. The Pythagorean theorem and the Vertical Angles Theorem are famous examples.
A theorem is built from definitions, postulates and earlier theorems by a proof. Its “if” part is the hypothesis and its “then” part is the conclusion.
Formal definition and notation
A theorem is a statement, usually of the form “if P, then Q“, that follows by valid logical deduction from axioms (postulates), definitions and previously proved theorems. The argument that establishes it is a proof. Before it is proved, such a statement is a conjecture; a small theorem used to prove a bigger one is a lemma, and a result that follows quickly from a theorem is a corollary.
Symbol
Read it as
Meaning
P ⇒ Q
P implies Q; if P, then Q
the logical form of most theorems: hypothesis P, conclusion Q
P ⇔ Q
P if and only if Q
a theorem whose converse is also true (a biconditional)
∴
therefore
introduces a conclusion inside a proof; the end of a proof is often marked Q.E.D. or a small square
Key facts
A postulate (axiom) is accepted without proof; a theorem must be proved. “Through two points there is exactly one line” is a postulate.
Most theorems have a hypothesis (the “if” part) and a conclusion (the “then” part). The conclusion is only guaranteed when the hypothesis is true.
The converse of a theorem swaps the parts. It may be true (the converse of the Pythagorean theorem) or false (“if two angles are equal, they are vertical angles” is false).
Checking many examples does not prove a theorem. One counterexample is enough to show a statement is false.
Common theorems in school math: the Pythagorean theorem, the Triangle Angle Sum Theorem (180 degrees), the Vertical Angles Theorem and the distributive property for numbers.
Try it: test a theorem
Interactive
The Pythagorean theorem says every right triangle has a2 + b2 = c2. Change the legs and check that it holds every time; that is what a proved theorem guarantees.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Answer:x = 20, and each vertical angle measures 70 degrees.
Example 3: Use a converse
Test-style
A triangle has sides 7, 24 and 25 centimeters. Is it a right triangle? Which theorem tells you?
The converse of the Pythagorean theorem: if a2 + b2 = c2 for the three sides, the triangle is a right triangle.
Use the longest side as c: 72 + 242 = 49 + 576 = 625.
252 = 625. The two sides match.
So the angle opposite the 25-centimeter side is a right angle.
Answer: Yes. By the converse of the Pythagorean theorem, the 7-24-25 triangle is a right triangle.
Common mistakes
✗ Mistake“I checked the formula on 10 triangles, so I proved it.”
✓ InsteadExamples can suggest a theorem, but only a general proof that covers every case proves it.
A pattern can hold for many cases and then fail. For example, n2 + n + 41 is prime for n = 0 to 39 but not for n = 40.
✗ MistakeUsing a theorem’s converse as if it were automatically true: “angles 2 and 5 are equal, so they are vertical angles.”
✓ InsteadCheck the converse separately. Equal angles need not be vertical angles.
Swapping the hypothesis and conclusion makes a new statement, which needs its own proof and may be false.
✗ MistakeApplying a2 + b2 = c2 to a triangle with no right angle.
✓ InsteadUse the Pythagorean theorem only when the hypothesis holds: the triangle is a right triangle.
A theorem guarantees its conclusion only when its hypothesis is true. For other triangles you need a different theorem, such as the law of cosines.
Not to be confused with
Theoremvs.Postulate (axiom)
A postulate is accepted as true without proof, such as “through two points there is exactly one line”. A theorem must be proved, often using postulates.
A formula is an equation for calculating something, such as A = π r2. A theorem is a proved statement; some theorems give formulas, such as a2 + b2 = c2.
Theoremvs.Conjecture
A conjecture is a statement believed to be true but not yet proved. Once someone proves it, it becomes a theorem.
Quick check: 3 questions
Question 1 of 3
What makes a statement a theorem?
Show answer and explanation
Answer: C. A theorem is a statement that has been proved. A statement accepted without proof is a postulate, one that works for some examples is only a conjecture, and many theorems contain no formula at all.
Question 2 of 3
In the theorem “If a triangle has two equal sides, then the angles opposite them are equal”, what is the hypothesis?
Show answer and explanation
Answer: B. The hypothesis is the “if” part: a triangle has two equal sides. The “then” part, that the opposite angles are equal, is the conclusion. The other choices are not part of the statement.
Question 3 of 3
Which side lengths make a right triangle, according to the converse of the Pythagorean theorem?
Show answer and explanation
Answer: D.92 + 122 = 81 + 144 = 225 = 152. For the other sets the sum of the squares of the two shorter sides does not equal the square of the longest side.
Theorem: frequently asked questions
What is the difference between a theorem and a postulate?
A postulate, also called an axiom, is a basic statement accepted as true without proof, such as “a line contains at least two points”. A theorem is a statement that has been proved from postulates, definitions and other theorems. Geometry starts with a few postulates and builds every theorem on top of them.
What are lemmas and corollaries?
Both are kinds of theorems. A lemma is a smaller result proved mainly to help prove a bigger theorem. A corollary is a result that follows quickly from a theorem that was just proved. For example, from the Triangle Angle Sum Theorem (angles add to 180 degrees) comes the corollary that the two acute angles of a right triangle add to 90 degrees.
Can a theorem ever be wrong?
A correctly proved theorem is true for every case its hypothesis covers, and it stays true. What can go wrong is using it outside its hypothesis, such as applying the Pythagorean theorem to a triangle without a right angle, or finding a mistake in a claimed proof, in which case the statement was never really a theorem.
Pythagorean TheoremThe Pythagorean theorem says that in a right triangle, a2 + b2 = c2. The legs are a and b, the two sides that form the right angle. The hypotenuse, c, is the longest side across from the right angle. Legs of 3 and 4 give 9 + 16 = 25, so the hypotenuse is 5.Geometry · Grade 8, Algebra 1ProofA proof is a step-by-step logical argument that shows a math statement is true in every case it covers. It starts from given facts and uses definitions, postulates and statements already proved, with a reason for each step. A statement that has been proved is called a theorem.Geometry · Geometry, Algebra 1SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3Vertical AnglesVertical angles are the pair of opposite angles formed when two lines intersect. They share a vertex but no sides, and they are always equal in measure. Two crossing lines make two pairs of vertical angles: if one angle is 120°, the angle across from it is also 120°.Geometry · Grade 7, Grade 8, GeometryCounterexampleA counterexample is a single example that shows a general statement is false. It fits all the conditions of the statement but does not have the claimed result. To disprove ‘every prime number is odd,’ one counterexample is enough: 2 is prime and even. Examples that agree with a claim never prove it, but one counterexample disproves it.Geometry · Grade 8, Algebra 1, GeometryTriangleA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).Geometry · Grade 3, Grade 4, Grade 5DegreeA degree is the standard unit for measuring an angle, written with the symbol °. One degree is 1360 of a full turn, so a full turn is 360°, a straight angle is 180° and a right angle is 90°. An angle that turns through 25 one-degree angles measures 25°.Geometry · Grade 4, Grade 5, Grade 7Distributive PropertyThe distributive property says that multiplying a number by a sum gives the same result as multiplying it by each term and then adding: a(b + c) = ab + ac. It multiplies a factor by each term inside parentheses, so 3(x + 4) = 3x + 12. It works with subtraction too, and using it backward is factoring.Algebra · Grade 6, Grade 7, Grade 8NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Right TriangleA right triangle is a triangle with one right angle, an angle of exactly 90 degrees. The two sides that form the right angle are the legs, and the longest side, across from the right angle, is the hypotenuse. Its sides follow the Pythagorean theorem: a2 + b2 = c2.Geometry · Grade 4, Grade 6, Grade 8HypotenuseThe hypotenuse is the longest side of a right triangle. It is always across from the right angle, and the two shorter sides that form the right angle are the legs. In the Pythagorean theorem a2 + b2 = c2, the hypotenuse is c: in a 3-4-5 right triangle, it is the side of length 5.Geometry · Grade 8, Algebra 1SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2CentimeterA centimeter is a metric unit of length equal to one hundredth of a meter, about the width of a fingernail. Its symbol is cm. There are 100 centimeters in 1 meter and 10 millimeters in 1 centimeter, and 1 inch equals 2.54 centimeters.Measurement · Grade 3, Grade 4, Grade 5Right AngleA right angle is an angle that measures exactly 90 degrees, a quarter of a full turn. It looks like the corner of a square or a sheet of paper, and it is marked with a small square at the vertex. Two lines that meet at a right angle are perpendicular.Geometry · Grade 2, Grade 3, Grade 4FormulaA formula is a rule written with symbols that shows how quantities are related. It is an equation with two or more variables, such as A = lw for the area of a rectangle. Formulas help solve problems the same way each time: substitute the values you know, then compute the one you need.Problem Solving · Grade 5, Grade 6, Grade 7AngleAn angle is formed by two rays that share the same endpoint, called the vertex. It is measured in degrees by how far one ray is turned from the other. A right angle measures 90 degrees, a straight angle measures 180 degrees, and one full turn is 360 degrees.Geometry · Grade 4, Grade 5, Grade 6Triangle Inequality TheoremThe triangle inequality theorem says that in any triangle, the sum of the lengths of any two sides is greater than the length of the third side. So 3, 4 and 5 can form a triangle, but 2, 3 and 6 cannot, because 2 + 3 < 6. Shortcut: the two shorter sides must add to more than the longest.Geometry · Grade 7, GeometryCongruent FiguresCongruent figures have exactly the same size and shape. One can be moved onto the other by sliding, flipping or turning, so every pair of matching sides and angles is equal. Two squares with side 5 are congruent; a 3-4-5 triangle and a 6-8-10 triangle are not, because their sizes differ. The symbol is ≅.Geometry · Grade 7, Grade 8Special Right TrianglesSpecial right triangles are the two right triangles whose angles are 45°-45°-90° or 30°-60°-90°. Their sides always come in fixed ratios: x : x : x√2 for the 45-45-90 triangle and x : x√3 : 2x for the 30-60-90 triangle. From one side you can find the other two without the Pythagorean theorem.Geometry · Grade 8, Geometry, Algebra 2EquationAn equation is a statement that two expressions are equal. It always has an equal sign, such as x + 5 = 12 or 3a = 18. Solving an equation means finding the value of the variable that makes it true: here x = 7 and a = 6. An expression like 4x + 7 has no equal sign, so it is not an equation.Algebra · Grade 6, Grade 7, Grade 8PlaneA plane is a perfectly flat surface that has no thickness and extends forever in every direction, like a tabletop that never ends. 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