Triangle congruence tests | Year 8

01 / LEARN

Understand the idea.

Enough information fixes a triangle

A congruence test gives enough information to fix a triangle's size and shape. SSS uses three corresponding equal sides. SAS uses two corresponding equal sides and their included angle: the angle where those two sides meet.

ASA uses two equal angles and the side between them; AAS uses two angles and another corresponding side. RHS applies to right-angled triangles with equal hypotenuses and one corresponding equal shorter side. The hypotenuse is always opposite the right angle.

Congruence tests
Test	Information needed
SSS	Three corresponding sides
SAS	Two sides and their included angle
ASA or AAS	Two angles and a corresponding side
RHS	Right angle, hypotenuse and a corresponding shorter side

Know what is not enough

Equal angles alone fix shape but allow different sizes. Triangles with sides $3, 4, 5$ and $6, 8, 10$ have matching angles, yet their lengths differ. AAA therefore establishes similarity rather than congruence.

Two equal side pairs and a non-included angle do not give a general congruence test: sometimes two different triangles satisfy those facts. Before claiming SAS, point to the common endpoint of the two given sides and check that its angle is supplied.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Recognise an included angle

In triangles ABC and DEF, AB = DE = 4 cm, AC = DF = 7 cm, and angles A and D both equal 50°. Prove congruence.

Worked solution

1
Record the two corresponding side pairs.
$A B = D E = 4 cm, A C = D F = 7 cm$
These pairings identify A with D, B with E and C with F.
2
Identify the included angle.
$∠ B A C = ∠ E D F = 50^{\circ}$
AB and AC meet at A; DE and DF meet at D, so each given angle lies between its two known sides.
3
Apply the matching congruence test.
$△ A B C ≅ △ D E F by SAS$
Two corresponding sides and the included angle are equal, which is sufficient for congruence.

Answer

Congruent by SAS.

EXAMPLE 02

Use a right angle

Two right-angled triangles each have hypotenuse 10 cm and a corresponding shorter side of 6 cm. Must they be congruent?

Worked solution

1
Check the right-angle condition.
$Each triangle has an angle of 90^{\circ}$
RHS can be used only for right-angled triangles.
2
Identify the equal side information.
$Hypotenuse = 10 cm, matching shorter side = 6 cm$
Both the hypotenuse and one corresponding shorter side have the same lengths.
3
Conclude congruence by RHS.
$Right angle + hypotenuse + side ⟹ congruent$
All three requirements of RHS are satisfied, so the triangles must be congruent.

Answer

Yes, by RHS.

03 / YOUR TURN

Try it yourself.

Check your understanding

Both triangles have side lengths 5 cm, 6 cm and 7 cm. Which congruence test applies?

Show answer & reasoning

SSS.

The matching side sets are $5 cm$ , $6 cm$ and $7 cm$ . All three corresponding sides are equal, so SSS establishes congruence.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Triangles have corresponding side lengths 4, 6, 8 cm and 4, 6, 8 cm. State the congruence test.
1 mark
Two triangles each have angles 40° and 60°, and the side between these angles is 7 cm in both. State the test.
2 marks
Two right triangles have hypotenuse 13 cm and a corresponding leg of 5 cm. State the test.
2 marks
Triangles each have angles 35°, 65° and 80°. Is congruence guaranteed? Explain.
2 marks
$A B = D E$ , $A C = D F$ and angles $A$ and $D$ are equal. State a test and identify why it applies.
3 marks
A student uses SSA as a general congruence test. Explain why this is unsafe.
2 marks

Read the worked solutions

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

Triangles have corresponding side lengths 4, 6, 8 cm and 4, 6, 8 cm. State the congruence test.
1 mark
All three side pairs match.
SSS.
Two triangles each have angles 40° and 60°, and the side between these angles is 7 cm in both. State the test.
2 marks
Two angles and their included side match.
ASA.
Two right triangles have hypotenuse 13 cm and a corresponding leg of 5 cm. State the test.
2 marks
Right angle, hypotenuse and one matching shorter side are equal.
RHS.
Triangles each have angles 35°, 65° and 80°. Is congruence guaranteed? Explain.
2 marks
AAA gives similarity; one triangle could be an enlargement of the other.
No. Equal angles do not fix the triangle’s size.
$A B = D E$ , $A C = D F$ and angles $A$ and $D$ are equal. State a test and identify why it applies.
3 marks
$A B$ and $A C$ meet at $A$ .
$D E$ and $D F$ meet at $D$ .
SAS, because $A$ and $D$ are included between the given side pairs.
A student uses SSA as a general congruence test. Explain why this is unsafe.
2 marks
The included-angle requirement of SAS has not been met.
Extra information or a valid test is needed.
A non-included angle with two sides may allow more than one triangle.

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum

Congruent shapes and matching parts; triangle congruence tests; similar shapes and corresponding sides; triangle similarity; properties of quadrilaterals established using triangles; testing congruence/similarity with transformations; decision-tree algorithms and digital geometry.

Official curriculum source ↗

In this lesson

Understand the idea.

Enough information fixes a triangle

Know what is not enough

See the method in action.

Recognise an included angle

Record the two corresponding side pairs.

Identify the included angle.

Apply the matching congruence test.

Use a right angle

Check the right-angle condition.

Identify the equal side information.

Conclude congruence by RHS.

Try it yourself.

Check your understanding

Keep practising