MathematicsChapter 9: Geometry and Polygons
Steps of A Proof
A geometry proof is easier when you always write the same headings.
Learning objectives
- Explain steps of a proof 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.
Definition
A geometry proof is easier when you always write the same headings. Start with the statement (the claim in words). Draw a figure that matches the statement, with letters. List the given facts. Write to prove. Add a construction only if you need an extra line, arc, or point. Then write the reasons, line by line, until the to-prove appears. Missing one of these steps is how a correct idea becomes an incomplete answer.
Where / symbols
- statement
- Start with the statement (the claim in words).
- figure
- Draw a figure that matches the statement, with letters.
- given
- List the given facts.
- to prove
- Write to prove.
- construction
- Add a construction only if you need an extra line, arc, or point.
- reasons
- Then write the reasons, line by line, until the to-prove appears.
Overview
Split the page into what you are allowed to use and what you still have to reach:
Statement Figure Given
The statement is the theorem or exercise in one clear sentence. The figure is a labelled sketch, not a work of art: enough to show the points you will name. Given is the hypothesis copied into symbols (AB = AC, angle P = 90 deg, and so on). If a fact is not in Given and not an axiom, you may not use it yet.
Examples Statement: in an isosceles triangle the base angles are equal · Figure: triangle ABC with AB = AC marked · Given: AB = AC
To Prove Construction Reasons
To prove is the conclusion in symbols (angle B = angle C). Construction is optional extra drawing: a bisector, a parallel, or a copied length. Reasons are the engine: each new line cites a definition, postulate, axiom, or earlier theorem. 'From the figure' is not a reason. Stop when the to-prove line is reached, not when the drawing looks finished.
Examples To prove: angle ABC = angle ACB · Construction: draw the bisector of angle A to meet BC · Reason: corresponding parts of congruent triangles (after SAS or SSS)
WHAT — Core idea
A geometry proof is easier when you always write the same headings. Start with the statement (the claim in words). Draw a figure that matches the statement, with letters. List the given facts. Write to prove. Add a construction only if you need an extra line, arc, or point. Then write the reasons, line by line, until the to-prove appears. Missing one of these steps is how a correct idea becomes an incomplete answer.
HOW — Rules, method, and examples
1. STATEMENT FIGURE GIVEN: The statement is the theorem or exercise in one clear sentence. The figure is a labelled sketch, not a work of art: enough to show the points you will name. Given is the hypothesis copied into symbols (AB = AC, angle P = 90 deg, and so on). If a fact is not in Given and not an axiom, you may not use it yet. 2. TO PROVE CONSTRUCTION REASONS: To prove is the conclusion in symbols (angle B = angle C). Construction is optional extra drawing: a bisector, a parallel, or a copied length. Reasons are the engine: each new line cites a definition, postulate, axiom, or earlier theorem. 'From the figure' is not a reason. Stop when the to-prove line is reached, not when the drawing looks finished.
WHY — Meaning and connections
Keep Given and To prove in different boxes so you cannot accidentally assume the answer. Construction comes before reasons if you need a helper line. A proof that never names a reason is only a story about the figure.
HOW — Answer in an exam
The six headings are statement, figure, given, to prove, construction, reasons. Do not hide the conclusion inside Given. Do not skip reasons and point at the sketch. Construction is only for helper lines you actually use.
Explanation
Keep Given and To prove in different boxes so you cannot accidentally assume the answer. Construction comes before reasons if you need a helper line. A proof that never names a reason is only a story about the figure.
In simple words
The six headings are statement, figure, given, to prove, construction, reasons. Do not hide the conclusion inside Given. Do not skip reasons and point at the sketch. Construction is only for helper lines you actually use.
Key points
- statement
- figure
- given
- to prove
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Steps of A Proof
Sticky Notes · Quick Revision
WHAT
- A geometry proof is easier when you always write the same headings.
Key Points
- A geometry proof is easier when you always write the same headings.
- The statement is the theorem or exercise in one clear sentence.
- To prove is the conclusion in symbols (angle B = angle C).
- Keep Given and To prove in different boxes so you cannot accidentally assume the answer.
HOW
- 1.
- The statement is the theorem or exercise in one clear sentence.
- To prove is the conclusion in symbols (angle B = angle C).
Examples
- Statement: in an isosceles triangle the base angles are equal
- Figure: triangle ABC with AB = AC marked
- Given: AB = AC
- To prove: angle ABC = angle ACB
- Construction: draw the bisector of angle A to meet BC
Common Mistakes
- The six headings are statement, figure, given, to prove, construction, reasons.
- Do not hide the conclusion inside Given.
- Do not skip reasons and point at the sketch.
Exam Tip
- The six headings are statement, figure, given, to prove, construction, reasons. Do not hide the conclusion inside Given. Do not skip reasons and point at the sketch. Construction is only for helper lines you actually use.
60-second revision
- A geometry proof is easier when you always write the same headings.
- The statement is the theorem or exercise in one clear sentence.
- To prove is the conclusion in symbols (angle B = angle C).
- Keep Given and To prove in different boxes so you cannot accidentally assume the answer.
- The six headings are statement, figure, given, to prove, construction, reasons.
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