MathematicsChapter 9: Geometry and Polygons
Statements and Propositions
Geometry is built from sentences you can judge.
Learning objectives
- Explain statements and propositions 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.
Statement
Geometry is built from sentences you can judge. A statement is a sentence that is true or false, such as 'a square has four equal sides'. An open sentence is not yet true or false, because a blank or a variable is still waiting, such as 'x is even'. You cannot prove an open sentence until you fill it. A proof is a stack of statements, each with a reason.
Where / symbols
- true or false
- A statement is a sentence that is true or false, such as 'a square has four equal sides'.
- open sentence
- An open sentence is not yet true or false, because a blank or a variable is still waiting, such as 'x is even'.
- not yet
- An open sentence is not yet true or false, because a blank or a variable is still waiting, such as 'x is even'.
Overview
Hold the parts of a claim apart so the proof does not mix them:
A Proposition
A proposition is a claim set out so that other people can follow it. It has a premise (what you start from), an argument (the chain of reasons), and a conclusion (what you reach). If the premises are accepted and the argument is valid, the conclusion must follow. A geometry theorem is a proposition that has already been proved.
Examples Premise: AB = AC. Argument: base angles of an isosceles triangle. Conclusion: angle B = angle C · Premise: a figure is a rectangle. Conclusion: its angles are 90 deg · A missing argument is just an opinion, not a proposition ready for class
Hypothesis and Conclusion
In a typical 'if ... then ...' line, the hypothesis is the given part (the if) and the conclusion is the to-prove part (the then). Mixing them is the most common way to start the wrong proof. Write the hypothesis under Given and the conclusion under To prove before you add any construction.
Examples If a triangle is isosceles, then two of its angles are equal · Hypothesis: PQRS is a parallelogram. Conclusion: opposite sides are equal · Swapping if and then writes a different claim (the converse)
WHAT — Core idea
Geometry is built from sentences you can judge. A statement is a sentence that is true or false, such as 'a square has four equal sides'. An open sentence is not yet true or false, because a blank or a variable is still waiting, such as 'x is even'. You cannot prove an open sentence until you fill it. A proof is a stack of statements, each with a reason.
HOW — Rules, method, and examples
1. A PROPOSITION: A proposition is a claim set out so that other people can follow it. It has a premise (what you start from), an argument (the chain of reasons), and a conclusion (what you reach). If the premises are accepted and the argument is valid, the conclusion must follow. A geometry theorem is a proposition that has already been proved. 2. HYPOTHESIS AND CONCLUSION: In a typical 'if ... then ...' line, the hypothesis is the given part (the if) and the conclusion is the to-prove part (the then). Mixing them is the most common way to start the wrong proof. Write the hypothesis under Given and the conclusion under To prove before you add any construction.
WHY — Meaning and connections
True and false belong to statements. Open sentences wait for a value: 'n is a multiple of 4' becomes a statement only when n is named. In a proof, every line after Given should be a statement you can defend, not a wish about the figure.
HOW — Answer in an exam
Do not treat an open sentence as a theorem. A proposition needs premise, argument, and conclusion. Hypothesis is the given; conclusion is what you must prove. Swapping them is a new claim, not the same proof.
Explanation
True and false belong to statements. Open sentences wait for a value: 'n is a multiple of 4' becomes a statement only when n is named. In a proof, every line after Given should be a statement you can defend, not a wish about the figure.
In simple words
Do not treat an open sentence as a theorem. A proposition needs premise, argument, and conclusion. Hypothesis is the given; conclusion is what you must prove. Swapping them is a new claim, not the same proof.
Key points
- true or false
- open sentence
- not yet
- premise
- argument
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Statements and Propositions
Sticky Notes · Quick Revision
WHAT
- Geometry is built from sentences you can judge.
Key Points
- Geometry is built from sentences you can judge.
- A proposition is a claim set out so that other people can follow it.
- In a typical 'if ... then ...' line, the hypothesis is the given part (the if) and the conclusion is the to-prove part (the then).
- True and false belong to statements.
HOW
- 1.
- A proposition is a claim set out so that other people can follow it.
- In a typical 'if ... then ...' line, the hypothesis is the given part (the if) and the conclusion is the to-prove part (the then).
Examples
- Premise: AB = AC. Argument: base angles of an isosceles triangle. Conclusion: angle B = angle C
- Premise: a figure is a rectangle. Conclusion: its angles are 90 deg
- A missing argument is just an opinion, not a proposition ready for class
- If a triangle is isosceles, then two of its angles are equal
- Hypothesis: PQRS is a parallelogram. Conclusion: opposite sides are equal
Common Mistakes
- Do not treat an open sentence as a theorem.
- A proposition needs premise, argument, and conclusion.
- Correction: Hypothesis is the given; conclusion is what you must prove.
Exam Tip
- Do not treat an open sentence as a theorem. A proposition needs premise, argument, and conclusion. Hypothesis is the given; conclusion is what you must prove. Swapping them is a new claim, not the same proof.
60-second revision
- Geometry is built from sentences you can judge.
- A proposition is a claim set out so that other people can follow it.
- In a typical 'if ... then ...' line, the hypothesis is the given part (the if) and the conclusion is the to-prove part (the then).
- True and false belong to statements.
- Do not treat an open sentence as a theorem.
Did you understand this topic?
Your answer is saved with your learning record. It does not mark the topic as mastered.