Congruent & Similar Triangles (Grade 10)

The four conditions for congruency (SSS, SAS, AAS and RHS), what similarity means, and how to set out a proof so that every statement earns its reason.

1Congruent versus similar

Congruent (≡) — same shape AND same size. One is an exact copy of the other.
Similar (|||) — same shape, different size. Angles equal, sides in proportion.

△ABC ≡ △DEF: each pair of matching sides carries the same number of ticks.

2The four congruency conditions

SSS — three pairs of equal sides
SAS — two sides and the angle BETWEEN them
AAS — two angles and a corresponding side
RHS — right angle, hypotenuse and one side

⚠️

There is no such condition as SSA. Two sides and a non-included angle can produce two different triangles, so it proves nothing. In SAS the angle must sit between the two sides.

💡

AAA is similarity, not congruency. Three equal angles fix the shape but say nothing about the size — that is exactly the difference between the two ideas.

3Setting out a congruency proof

Worked ExampleIn △ABC and △DEF, AB = DE, BC = EF and CA = FD. Prove the triangles are congruent.

1
AB = DE (given)
Every statement needs a reason in brackets.
2
BC = EF (given)
3
CA = FD (given)
4
∴ △ABC ≡ △DEF (SSS)
Name the condition — that is the final mark.

Write the letters in matching order. △ABC ≡ △DEF says A matches D, B matches E and C matches F. Getting the order wrong loses marks even when the reasoning is right.

4Similarity

If △ABC ||| △DEF then

Â = D̂, B̂ = Ê, Ĉ = F̂ and ABDE = BCEF = CAFD

Worked Example△ABC ||| △DEF with AB = 6, DE = 9 and BC = 8. Calculate EF.

1
ABDE = BCEF.
Corresponding sides are in proportion.
2
69 = 8EF.
Substitute.
3
6 × EF = 72.
Cross-multiply.
4
EF = 12.
Check: 69 = 812 = 23. ✓

💡

Build every proportion as small trianglebig triangle = small trianglebig triangle. Keeping the same triangle on top throughout stops the fraction being flipped halfway.

Practice exercises

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

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
State the four conditions for congruency.
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SSS, SAS, AAS and RHS
2
In △ABC and △DEF, AB = DE, BC = EF and CA = FD. Prove that the triangles are congruent.
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All three pairs of sides are equal (given), so △ABC ≡ △DEF (SSS)
3
Explain why SSA is not a valid congruency condition.
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Two sides with a non-included angle can produce two different triangles, so it does not fix the triangle.
4
What is the difference between congruent and similar triangles?
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Congruent = same shape and same size; similar = same shape, sides in proportion.
5
△ABC ||| △DEF with AB = 6, DE = 9 and BC = 8. Calculate EF.
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69 = 8EF → EF = 12
6
Two triangles have all three angles equal. Are they congruent?
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No — they are similar. Equal angles fix the shape but not the size.
7
Which condition uses a right angle, the hypotenuse and one other side?
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RHS
8
In SAS, where must the angle be?
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Between the two given sides (the included angle).