Theorem and Converse

Theorem and Converse

What is it?

After axioms come the claims you must work for. A conjecture is a guess that still needs proof. A theorem is a statement that can be proved from axioms, postulates, definitions, and earlier theorems. A corollary is a theorem that follows so quickly from another that it needs almost no extra argument. The converse of a statement interchanges the given and the to-prove. The converse need not be true, so it needs its own check.

In simple words

• After axioms come the claims you must work for.
• A conjecture still needs proof: 'I think every even number greater than 2 is a sum of two primes' is a famous guess, not a Class 9 theorem.
• To write the converse, interchange the given and the to-prove. 'If a triangle is equilateral, then it is equiangular' has converse 'if a triangle is equiangular, then it is equilateral'.
• Exam papers love the word converse because students copy the original proof and think they have proved the swap.

Important terms

conjecture: A conjecture is a guess that still needs proof.

theorem: A theorem is a statement that can be proved from axioms, postulates, definitions, and earlier theorems.

corollary: A corollary is a theorem that follows so quickly from another that it needs almost no extra argument.

converse: The converse of a statement interchanges the given and the to-prove.

need not be true: The converse need not be true, so it needs its own check.

What you should learn

• Explain theorem and converse in clear mathematical language.
• Use the main rule, formula, test, or construction correctly.
• Work through a simple example and show each important step.
• Identify common mistakes and write an exam-ready conclusion.

Examples

• Guess from a sketch: the diagonals look equal (still a conjecture)
• Proved: opposite sides of a parallelogram are equal (a theorem)
• From triangle sum 180 deg, a right triangle's acute angles add to 90 deg (a corollary)

CONJECTURE TO THEOREM

A conjecture still needs proof: 'I think every even number greater than 2 is a sum of two primes' is a famous guess, not a Class 9 theorem. A theorem can be proved from axioms and earlier results, and then you may quote it. A corollary follows quickly: if you have proved the angle sum of a triangle is 180 deg, then each angle of an equilateral triangle is 60 deg.

THE CONVERSE

To write the converse, interchange the given and the to-prove. 'If a triangle is equilateral, then it is equiangular' has converse 'if a triangle is equiangular, then it is equilateral'. That pair happens to be true, but many converses fail. 'If a figure is a square, then it is a rectangle' is true; the converse 'if a figure is a rectangle, then it is a square' is false.

Full school-level explanation

After axioms come the claims you must work for. A conjecture is a guess that still needs proof. A theorem is a statement that can be proved from axioms, postulates, definitions, and earlier theorems. A corollary is a theorem that follows so quickly from another that it needs almost no extra argument. The converse of a statement interchanges the given and the to-prove. The converse need not be true, so it needs its own check.

Name the four words so you do not promote a guess to a theorem too soon:

WHAT — Core idea

After axioms come the claims you must work for. A conjecture is a guess that still needs proof. A theorem is a statement that can be proved from axioms, postulates, definitions, and earlier theorems. A corollary is a theorem that follows so quickly from another that it needs almost no extra argument. The converse of a statement interchanges the given and the to-prove. The converse need not be true, so it needs its own check.

HOW — Rules, method, and examples

1. CONJECTURE TO THEOREM: A conjecture still needs proof: 'I think every even number greater than 2 is a sum of two primes' is a famous guess, not a Class 9 theorem. A theorem can be proved from axioms and earlier results, and then you may quote it. A corollary follows quickly: if you have proved the angle sum of a triangle is 180 deg, then each angle of an equilateral triangle is 60 deg. 2. THE CONVERSE: To write the converse, interchange the given and the to-prove. 'If a triangle is equilateral, then it is equiangular' has converse 'if a triangle is equiangular, then it is equilateral'. That pair happens to be true, but many converses fail. 'If a figure is a square, then it is a rectangle' is true; the converse 'if a figure is a rectangle, then it is a square' is false.

WHY — Meaning and connections

Exam papers love the word converse because students copy the original proof and think they have proved the swap. You have not. Treat the converse as a new proposition: new Given, new To prove. If a counter-example exists, stop and say the converse is false.

HOW — Answer in an exam

A conjecture is not a theorem until it is proved. A corollary is a fast follow-on, not a new deep proof. The converse swaps given and to-prove and need not be true. Never assume the converse of a theorem without checking.

CONJECTURE TO THEOREM

A conjecture still needs proof: 'I think every even number greater than 2 is a sum of two primes' is a famous guess, not a Class 9 theorem. A theorem can be proved from axioms and earlier results, and then you may quote it. A corollary follows quickly: if you have proved the angle sum of a triangle is 180 deg, then each angle of an equilateral triangle is 60 deg.

Remember

• needs proof
• can be proved
• follows quickly

Examples

• Guess from a sketch: the diagonals look equal (still a conjecture)
• Proved: opposite sides of a parallelogram are equal (a theorem)
• From triangle sum 180 deg, a right triangle's acute angles add to 90 deg (a corollary)

THE CONVERSE

To write the converse, interchange the given and the to-prove. 'If a triangle is equilateral, then it is equiangular' has converse 'if a triangle is equiangular, then it is equilateral'. That pair happens to be true, but many converses fail. 'If a figure is a square, then it is a rectangle' is true; the converse 'if a figure is a rectangle, then it is a square' is false.

Remember

• interchange
• given
• to-prove

Examples

• Statement: if ABCD is a square, then its sides are equal. Converse is not automatic
• True converse pair: equilateral triangle <=> equiangular triangle
• False converse: rectangle => square

Remember

• conjecture
• theorem
• corollary
• converse
• need not be true
• new proposition
• counter-example
• not a theorem until it is proved

How the ideas connect

Exam papers love the word converse because students copy the original proof and think they have proved the swap. You have not. Treat the converse as a new proposition: new Given, new To prove. If a counter-example exists, stop and say the converse is false.

Deeper understanding

Go beyond memorising Theorem and Converse. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

Exam papers love the word converse because students copy the original proof and think they have proved the swap. You have not. Treat the converse as a new proposition: new Given, new To prove. If a counter-example exists, stop and say the converse is false.

Common mistakes

Exam focus

WHAT

Key Points

HOW

Examples

Exam focus

Quick revision

• conjecture
• theorem
• corollary
• converse
• conjecture: A conjecture is a guess that still needs proof.
• theorem: A theorem is a statement that can be proved from axioms, postulates, definitions, and earlier theore

A conjecture is not a theorem until it is proved. A corollary is a fast follow-on, not a new deep proof. The converse swaps given and to-prove and need not be true. Never assume the converse of a theorem without checking.

For long answers on Theorem and Converse: define clearly → explain the main parts → give one example → state one exam caution.