Proportionality & Similarity Theorems (Grade 12)

The proportion theorem for a line parallel to a side of a triangle, similar triangles and the AAA condition, and how to lay out a proportion so the ratios cannot be flipped.

1The proportion theorem

A line drawn parallel to one side of a triangle divides the other two sides proportionally.

DE ∥ BC, so ADDB = AEEC.

The reason to quote in an exam is (line ∥ one side of △) — or 'prop theorem; DE ∥ BC'. Without the reason the statement earns nothing.

💡

Set every proportion out the same way round: top piece over bottom piece on both sides. Mixing ADDB with ECAE is the commonest error in the whole topic.

2Similar triangles

△ABC ||| △DEF means
Â = D̂, B̂ = Ê, Ĉ = F̂ and ABDE = BCEF = ACDF

In Grade 12 the condition you use is AAA (equiangular). If two triangles have all three angles equal, their sides are in proportion — and you only need to prove two pairs of angles equal, since the third follows automatically.

Worked ExampleIn △ABC, DE ∥ BC. Prove that △ADE ||| △ABC.

1
Â is common to both triangles.
Always look for the shared angle first.
2
AD̂E = AB̂C (corresponding ∠s; DE ∥ BC)
Parallel lines give equal corresponding angles.
3
AÊD = AĈB (corresponding ∠s; DE ∥ BC)
4
∴ △ADE ||| △ABC (AAA)
Write the letters in matching order — A with A, D with B, E with C.

3Using similarity to find a length

Worked ExampleDE ∥ BC with AD = 4, DB = 6 and AE = 5. Calculate EC.

1
ADDB = AEEC.
Proportion theorem.
2
46 = 5EC.
Substitute.
3
4 × EC = 30.
Cross-multiply.
4
EC = 7,5.
Check: 46 = 57,5 = 23. ✓

⚠️

AD = 4 and DB = 6 means AB = 10, not 6. Read carefully whether a question gives you the piece or the whole side — ADAB = 410 is a different ratio from ADDB = 46.

4Proving Pythagoras with similar triangles

The ATP asks you to prove the Theorem of Pythagoras using similarity. Drop a perpendicular from the right angle to the hypotenuse and three similar triangles appear at once.

△ABC with Â = 90° and AD ⊥ BC. This single line creates three similar triangles.

Worked ExampleProve that AB² + AC² = BC²

1
In △ABD and △CBA: B̂ is common, and AD̂B = BÂC = 90°.
Two pairs of equal angles.
2
∴ △ABD ||| △CBA (AAA), so ABCB = BDBA.
Corresponding sides in proportion.
3
∴ AB² = BD · BC.
Cross-multiply.
4
Similarly △ACD ||| △BCA gives AC² = CD · CB.
The same argument on the other side.
5
Add: AB² + AC² = BC(BD + CD).
Take BC out as a common factor.
6
But BD + CD = BC, so AB² + AC² = BC². □
D lies on BC, so the two pieces make the whole.

💡

Check it on the 3-4-5 triangle above: BD = 1,80, CD = 3,20, so BD × BC = 1,80 × 5 = 9 = 3² = AB². ✓

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
In △ABC below, DE ∥ BC. Write down the proportion that follows, with a reason.
Show answer ▾Hide answer ▴
ADDB = AEEC (line ∥ one side of △)
2
DE ∥ BC with AD = 4, DB = 6 and AE = 5. Calculate EC.
Show answer ▾Hide answer ▴
46 = 5EC → EC = 7,5
3
In △ABC below with DE ∥ BC, prove that △ADE ||| △ABC.
Show answer ▾Hide answer ▴
Â common; AD̂E = AB̂C and AÊD = AĈB (corresponding ∠s, DE ∥ BC) → △ADE ||| △ABC (AAA)
4
If AD = 4 and DB = 6, write down the length of AB.
Show answer ▾Hide answer ▴
4 + 6 = 10
5
How many pairs of equal angles must you prove for AAA?
Show answer ▾Hide answer ▴
Two — the third follows because the angles of a triangle add to 180°.
6
△ABC ||| △DEF with AB = 8, DE = 12 and BC = 10. Calculate EF.
Show answer ▾Hide answer ▴
812 = 10EF → EF = 15
7
State the reason you must quote when using the proportion theorem.
Show answer ▾Hide answer ▴
(line ∥ one side of △)
8
Two triangles are equiangular. What can you conclude?
Show answer ▾Hide answer ▴
They are similar, so their corresponding sides are in proportion.