Grade 9 Axioms: Short Notes

9 ශ්‍රේණිය · 15 පරිච්ඡේදය · ප්‍රත්‍යක්ෂ

45%

of Grade 9 term tests include it (18 of 40)

marks in past papers (22 questions)

9 min

full lesson with quick checks

🎯 What you'll learn

Explain what an axiom is
State the axioms about equal quantities and the whole and its part
Use the axioms to show that line segments or angles are equal
Give the correct axiom as the reason for each step
Use the axioms to solve simple equations

Avurudu money

At Avurudu, Kasun and his cousin Dilini each get Rs 500 from their aunt. Then seeya gives each of them another Rs 200. Do they still have the same amount? Of course! Nobody needs to prove it. Facts this obvious are called axioms, and all of geometry is built on them.

What is an axiom?

An axiom is a statement we accept as true without proof, because it is clearly true.In geometry, we use axioms as reasons in proofs. Instead of measuring with a ruler or protractor, we explain *why* two lengths or angles must be equal.A picture can mislead; a reason cannot.

The axioms (1 to 3)

1. Quantities equal to the same quantity are equal to each other. (a = b, b = c, so a = c)2. If equals are added to equals, the results are equal. (a = b, so a + c = b + c)3. If equals are subtracted from equals, the remainders are equal. (a = b, so a − c = b − c)

The axioms (4 to 6)

4. If equals are multiplied by equals, the results are equal. (a = b, so 2a = 2b)5. If equals are divided by equal non-zero quantities, the results are equal. (a = b, so a/3 = b/3)6. The whole is greater than any of its parts. (AB > AC when C lies between A and B)

Take away the common part

AD (blue) = CB (amber), and both contain the common part CD (shaded). Subtract CD from both (axiom 3): AC = DB.

✏️ Adding the common part

A, C, D and B lie on a straight line in that order, and AC = DB. Show that AD = CB.

1AC = DB (given).
2Add CD to both: AC + CD = DB + CD (equals added to equals give equals).
3But AC + CD = AD and CD + DB = CB.
4So AD = CB.

✅ AD = CB (axiom 2)

⚠️ Add or subtract?

Ask: are you going from parts to wholes (bigger) or wholes to parts (smaller)?• Given the small pieces equal, want the big ones: add the common part (axiom 2).• Given the big ones equal, want the small pieces: subtract the common part (axiom 3).In papers, students lose marks for naming the wrong axiom, or for giving no reason at all.

✏️ Axioms with angles

In the figure, ∠AOC = ∠BOD, ∠AOB = 40° and ∠AOD = 150°. Find ∠COD and ∠BOC.

1∠AOC = ∠AOB + ∠BOC and ∠BOD = ∠BOC + ∠COD.
2∠AOC = ∠BOD, so subtract the common ∠BOC: ∠AOB = ∠COD (axiom 3).
3So ∠COD = 40°.
4∠BOC = ∠AOD − ∠AOB − ∠COD = 150° − 40° − 40° = 70°.

✅ ∠COD = 40°, ∠BOC = 70°

Axioms solve equations too

An equation says two quantities are equal, so the axioms let us do the same thing to both sides.3x + 5 = 20Subtract 5 from both sides (axiom 3): 3x = 15Divide both sides by 3 (axiom 5): x = 5Some questions ask you to name the axiom at each step.

⚠️ Don't trust the picture

Common wrong answers come from guessing: 'x = y because they look the same', or 'AOE is a straight line' when nothing says so. Figures are usually not drawn to scale. Use only what is given (or marked), and an axiom as the reason for every step.

📌 Remember

• An axiom is accepted as true without proof.• Equal to the same thing → equal to each other.• Add, subtract, multiply or divide (not by zero) equals by equals → results are equal.• The whole is greater than its part.• Find the common part: add it to go bigger, subtract it to go smaller, and name the axiom.

🧾 Quick revision sheet

What is an axiom?	A statement accepted as true without proof.
a = b and b = c. Then?	a = c: things equal to the same thing are equal.
a = b. Then a + c = ?	b + c: equals added to equals give equals.
a = b. Then a − c = ?	b − c: equals taken from equals leave equals.
The whole and its part	The whole is greater than any of its parts.
Big equal, want small pieces equal?	Subtract the common part (axiom 3).

Now test yourself: 5 quick checks and 1 more worked examples

The full lesson checks your answers as you go, and Gedara Sir tracks which parts you've got.

📖 Start the 9-minute lesson free ▶ Practise 5 past-paper questions