Congruent & Similar Triangles — Grade 9 Maths (CAPS)

The four conditions that prove two triangles congruent — SSS, SAS, AAS and RHS — how to write the correspondence correctly, and how similarity differs. With drawn figures for every condition, worked proofs and a quiz.

Also known as: congruent triangles · SSS SAS ASA RHS · similar triangles

In Grade 8 you learned what congruent and similar mean. Grade 9 asks something harder: prove it. You do not need to check all six measurements (three sides and three angles) — there are four shortcuts, and knowing which one to quote is where the marks are.

1Writing the correspondence correctly

When you write △ABC ≡ △DEF you are claiming that A matches D, B matches E and C matches F in that order. The order is not decoration — it tells the marker which sides and angles you are pairing.

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Writing △ABC ≡ △EDF when the vertices actually pair A→D, B→E, C→F is marked wrong, even if the triangles really are congruent. Always list the vertices in matching order.

2Condition 1: SSS (side, side, side)

If all three pairs of sides are equal, the triangles are congruent. Matching sides are shown with matching tick marks.

SSS: every pair of matching sides is equal, so the triangles must be congruent.

3Condition 2: SAS (side, angle, side)

Two pairs of sides are equal and the angle between them is equal. The angle must be the included angle — the one formed by those two sides.

SAS: the equal angle lies between the two marked sides.

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If the equal angle is not between the two equal sides, this is not SAS and does not prove congruency. Check that the angle sits between the two marked sides.

4Condition 3: AAS (angle, angle, side)

Two pairs of angles are equal and one pair of matching sides is equal. Once two angles match, the third must match too (angles in a triangle add to 180°) — so only one side is needed.

AAS: two equal angles plus one matching equal side.

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AAA is not a congruency condition. Three equal angles only make the triangles the same shape — that is similarity, not congruency. Without a side you cannot fix the size.

5Condition 4: RHS (right angle, hypotenuse, side)

For right-angled triangles only: if the hypotenuses are equal and one other pair of sides is equal, the triangles are congruent.

RHS: a right angle in each, equal hypotenuses and one other equal side.

6Writing a congruency proof

Worked ExampleGiven AB = DE, BC = EF and B̂ = Ê, prove that △ABC ≡ △DEF

1
Statement: AB = DE (given)
Quote each equal pair with its reason.
2
Statement: BC = EF (given)
3
Statement: B̂ = Ê (given)
This angle lies between the two equal sides.
4
Therefore △ABC ≡ △DEF (SAS)
Always end by naming the condition used — it carries its own mark.

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Lay proofs out as statement then reason, one pair per line, and finish with the condition (SSS / SAS / AAS / RHS). Markers award the reasons separately from the statements.

7Similarity: same shape, not necessarily the same size

Two triangles are similar (written |||) when their angles are equal and their sides are in the same ratio. For triangles, equal angles is enough — if two pairs of angles match, the triangles are similar.

△ABC ||| △DEF ⇒ AB/DE = BC/EF = AC/DF

The minimum conditions work the other way round as well. Two triangles are similar if either two pairs of corresponding angles are equal (the third pair then has to be too), or all three pairs of corresponding sides are in the same ratio.

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Two pairs of sides in the same ratio, on their own, do not prove similarity. 6 and 9, 8 and 12 are both in the ratio 1 : 1,5 — but until you know the third pair is in that ratio too (or that a pair of angles is equal), the triangles could still be different shapes. Check all three pairs, or work with the angles.

Worked Example△ABC ||| △DEF with AB = 4, DE = 12 and BC = 5. Find EF

1
Scale factor = DE ÷ AB = 12 ÷ 4 = 3.
Divide a side of the big triangle by its matching small side.
2
EF = BC × 3 = 5 × 3.
Every side is enlarged by the same factor.
3
EF = 15.
Check: 4 : 12 = 5 : 15 ✓

Practice exercises

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

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Name the four conditions for congruent triangles.
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SSS, SAS, AAS and RHS
2
Why is AAA not a congruency condition?
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Equal angles fix only the shape, not the size — that gives similarity, not congruency.
3
In SAS, where must the equal angle be?
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Between the two equal sides (the included angle).
4
Which condition applies only to right-angled triangles?
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RHS
5
If △ABC ≡ △PQR, which side equals BC?
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QR (B pairs with Q and C with R)
6
If △ABC ≡ △PQR, which angle equals Â?
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P̂
7
△ABC ||| △DEF with AB = 3, DE = 9 and BC = 4. Find EF.
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Scale factor 3 → EF = 12
8
The two triangles below are drawn to the same scale. Determine whether they are similar, giving a reason, and write down the scale factor from the smaller to the larger.
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9 ÷ 6 = 1,5; 12 ÷ 8 = 1,5; 15 ÷ 10 = 1,5 — all three pairs of corresponding sides are in the same ratio, so yes, the triangles are similar, with scale factor 1,5