Triangle similarity | Year 8

01 / LEARN

The idea, step by step

Three ways to establish the same shape

AA is enough for triangle similarity because if two pairs of angles match, the third pair must also match. A triangle’s angles sum to 180°.

Alternatively, prove all three corresponding side ratios equal, or prove two corresponding side ratios equal together with the included angle. Those tests establish proportional SSS and proportional SAS similarity. A ratio check must compare matching sides, not arbitrary pairs.

State a correspondence first

Write the triangles in matching vertex order. In $△ A B C \sim △ D E F$ , sides AB and DE correspond, as do AC and DF and BC and EF. Each ratio from ABC to DEF must be equal.

Congruent triangles are the special case of similar triangles with scale factor 1. Similarity alone does not let you set corresponding side lengths equal. It gives a common ratio.

A common trap

Saying that two triangles are similar because only one pair of side ratios matches.

Instead: Use AA or enough corresponding side/angle information to establish a valid similarity test.

02 / WORKED EXAMPLES

See the thinking

Example 1

Angles lead to a length

Triangles ABC and DEF have equal angles A and D, and equal angles B and E. AB = 8 cm, DE = 12 cm and AC = 10 cm. Find DF.

The triangles are similar by AA, with A matching D and B matching E.
The scale factor from ABC to DEF is $12 / 8 = 1.5$ .
DF corresponds to AC, so $D F = 10 (1.5) = 15$ cm.

15 cm.

Example 2

Check all three ratios

Triangle side lengths are 3, 4, 5 cm and 4.5, 6, 7.5 cm. Are the triangles similar?

Match shortest with shortest, middle with middle and longest with longest.
The ratios are $4.5 / 3 = 1.5$ , $6 / 4 = 1.5$ and $7.5 / 5 = 1.5$ .
All three match, so the triangles are similar by proportional SSS.

Yes; scale factor 1.5 from the first to the second.

Check your understanding

Two triangles each have angles 30° and 80°. Are they similar?

Show answer

Yes, by AA.

Their third angles are both 70°, so every corresponding angle matches.

03 / WORKSHEET

Your turn

6 questions · 15 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A triangle has angles 45° and 75°. Another has angles 45° and 60°. Are they similar? Explain.
3 marks
Are triangles with sides 4, 5, 6 cm and 8, 10, 12 cm similar? State a reason.
2 marks
Are triangles with sides 3, 4, 5 cm and 6, 8, 11 cm similar?
2 marks
Two triangles have included angle 50°. The sides enclosing it are 6 cm and 9 cm in one, 10 cm and 15 cm in the other. Are they similar?
3 marks
Given $△ A B C \sim △ D E F$ , AB = 5 cm, DE = 15 cm and BC = 7 cm. Find EF.
3 marks
Two similar triangles have scale factor 1. Explain what stronger conclusion follows.
2 marks

Read the worked solutions

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

A triangle has angles 45° and 75°. Another has angles 45° and 60°. Are they similar? Explain.
3 marks
The first triangle’s third angle is 60°.
Both have angle set 45°, 60°, 75°.
Yes, by AA.
Are triangles with sides 4, 5, 6 cm and 8, 10, 12 cm similar? State a reason.
2 marks
$8 / 4 = 10 / 5 = 12 / 6 = 2$ .
Yes, proportional SSS with factor 2.
Are triangles with sides 3, 4, 5 cm and 6, 8, 11 cm similar?
2 marks
Two ratios are 2, but $11 / 5 = 2.2$ .
All corresponding side ratios must match.
No.
Two triangles have included angle 50°. The sides enclosing it are 6 cm and 9 cm in one, 10 cm and 15 cm in the other. Are they similar?
3 marks
$10 / 6 = 15 / 9 = 5 / 3$ , and the included angles match.
Yes, by proportional SAS.
Given $△ A B C \sim △ D E F$ , AB = 5 cm, DE = 15 cm and BC = 7 cm. Find EF.
3 marks
Scale factor is $15 / 5 = 3$ .
$E F = 7 (3) = 21$ cm.
21 cm.
Two similar triangles have scale factor 1. Explain what stronger conclusion follows.
2 marks
Corresponding angles match by similarity, and factor 1 means all corresponding lengths are equal.
They are congruent.

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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 ↗

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