Congruent/similar triangles in proofs | Year 10

01 / LEARN

Understand the idea.

Sufficient information

Congruence means matching shape and size. SSS, SAS, ASA, AAS and RHS each supply sufficient information. In SAS, the angle must be between the two matching sides; RHS requires right triangles, equal hypotenuses and one equal corresponding leg.

AAA fixes shape but leaves size free, so it proves similarity. SSA is not a general congruence test: a non-included angle and two sides can sometimes fit two different triangles.

Use the result, not the target

Start a proof by listing the given equalities and any shared side. Choose a valid test, then state the triangle names in matching order. Only then use the equality of remaining corresponding sides or angles.

Do not include the target equality among your starting facts. If using similarity instead, deduce proportional sides first; lengths become equal only when the scale factor is $1$ .

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A shared diagonal

In quadrilateral ABCD, AB=AD and BC=CD. Prove ∠BAC=∠CAD.

Worked solution

1
Record the two given side pairs.
$A B = A D, B C = D C$
These are the stated conditions on the quadrilateral.
2
Include the common side.
$A C = A C$
Both triangles contain the same segment AC.
3
Prove the triangles congruent.
$△ A B C ≅ △ A D C$
The three side pairs match, so SSS congruence applies.
4
Match the angles at A.
$∠ B A C = ∠ C A D$
Congruence matches B with D and A with A, so the angles between AB and AC, and between AD and AC, are equal.

Answer

∠BAC=∠CAD

EXAMPLE 02

Lengths from similarity

Right triangles have corresponding legs 3, 4 and 6, 8. Explain why they are similar but not congruent.

Worked solution

1
Compare corresponding leg lengths.
$\frac{3}{6} = \frac{4}{8} = \frac{1}{2}$
The two pairs of sides are proportional.
2
Use the included right angles.
$90^{\circ} = 90^{\circ}$
The listed legs meet at a right angle in each triangle, so SAS similarity applies.
3
Check whether the sizes also match.
$k = \frac{6}{3} = \frac{8}{4} = 2 \neq 1$
Every larger length is twice its corresponding smaller length, which rules out congruence.

Answer

Similar, not congruent.

03 / YOUR TURN

Try it yourself.

Check your understanding

Two right triangles have equal hypotenuses and one equal corresponding leg. Which congruence test applies?

Show answer & reasoning

RHS

RHS applies: both triangles have a right angle, their hypotenuses are equal, and one pair of corresponding legs is equal. The hypotenuse must be the side opposite the right angle.

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6 worksheet questions · 15 marks

Open worksheet Worked solutions

Read the questions online

Two triangles have corresponding sides 5, 7, 8 and 5, 7, 8. Name the congruence test.
2 marks
Two triangles share side AC. $A B = A D$ and $B C = D C$ . Prove triangles ABC and ADC congruent.
3 marks
Explain why sides 5 and 7 with a non-included $30^{\circ}$ angle do not automatically prove congruence.
3 marks
Triangles ABC and DEF have $A \leftrightarrow D, B \leftrightarrow E, C \leftrightarrow F$ and are congruent. $∠ B = 52^{\circ}$ and $A C = 9$ cm. State $∠ E$ and DF.
2 marks
Two triangles have proportional sides $2 : 3, 4 : 6, 6 : 9$ . Can they be non-degenerate triangles? Explain before applying a similarity test.
3 marks
Two triangles have corresponding angles $70^{\circ}$ and $45^{\circ}$ , and the included corresponding side is 6 cm in each. Are they congruent?
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Two triangles have corresponding sides 5, 7, 8 and 5, 7, 8. Name the congruence test.
2 marks
All three pairs of corresponding sides match.
SSS
Two triangles share side AC. $A B = A D$ and $B C = D C$ . Prove triangles ABC and ADC congruent.
3 marks
$A B = A D$ and $B C = D C$ are given.
$A C = A C$ is shared.
SSS congruence.
Explain why sides 5 and 7 with a non-included $30^{\circ}$ angle do not automatically prove congruence.
3 marks
The angle is not included between the known sides.
An ambiguous second triangle may exist.
This is SSA, not a general congruence test.
Triangles ABC and DEF have $A \leftrightarrow D, B \leftrightarrow E, C \leftrightarrow F$ and are congruent. $∠ B = 52^{\circ}$ and $A C = 9$ cm. State $∠ E$ and DF.
2 marks
Use corresponding angles and sides after congruence.
$52^{\circ}$ and 9 cm
Two triangles have proportional sides $2 : 3, 4 : 6, 6 : 9$ . Can they be non-degenerate triangles? Explain before applying a similarity test.
3 marks
$2 + 4 = 6$ violates the strict triangle inequality.
The listed points would be collinear.
No.
Two triangles have corresponding angles $70^{\circ}$ and $45^{\circ}$ , and the included corresponding side is 6 cm in each. Are they congruent?
2 marks
Two angles and their included side match.
Yes, by ASA.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum

Angle and similarity reasoning; congruent/similar triangles in proofs; chords, radii and tangents; angles at the centre and circumference; circle theorems and their applications; connected deductive arguments; construction/spatial algorithms; testing and justifying solutions.

Official curriculum source ↗

In this lesson

Understand the idea.

Sufficient information

Use the result, not the target

See the method in action.

A shared diagonal

Record the two given side pairs.

Include the common side.

Prove the triangles congruent.

Match the angles at A.

Lengths from similarity

Compare corresponding leg lengths.

Use the included right angles.

Check whether the sizes also match.

Try it yourself.

Check your understanding

Keep practising