Triangle similarity | Year 8

01 / LEARN

Understand the idea.

Three ways to establish the same shape

For triangles, AA similarity needs two equal corresponding angle pairs. The third pair then follows from the $180^{\circ}$ angle sum. This fixes the shape, although the two triangles may have different sizes.

Other valid tests are proportional SSS, where all three corresponding side ratios agree, and proportional SAS, where two side ratios agree and the included angles are equal. A single matching ratio cannot establish similarity.

State a correspondence first

State the vertex correspondence before calculating. In $△ A B C \sim △ D E F$ , the pairs are $A \leftrightarrow D$ , $B \leftrightarrow E$ and $C \leftrightarrow F$ . Thus $A B$ matches $D E$ , $A C$ matches $D F$ and $B C$ matches $E F$ .

Choose one ratio direction and use it consistently. From $A B C$ to $D E F$ , $\frac{D E}{A B} = \frac{D F}{A C} = \frac{E F}{B C}$ . Corresponding sides are equal only when that common factor is 1; otherwise multiply by the factor to find the new length.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

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.

Worked solution

1
Establish the triangle correspondence.
$∠ A = ∠ D, ∠ B = ∠ E ⟹ △ A B C \sim △ D E F$
Two matching angle pairs establish AA similarity, with C corresponding to F.
2
Calculate the factor from ABC to DEF.
$k = \frac{D E}{A B} = \frac{12}{8} = 1.5$
DE corresponds to AB, so this ratio has the required direction.
3
Scale the side corresponding to AC.
$D F = k \times A C = 1.5 \times 10 = 15 cm$
Similar triangles have the same multiplier for each corresponding side pair.

Answer

15 cm.

EXAMPLE 02

Check all three ratios

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

Worked solution

1
Pair sides by their relative lengths.
$3 \leftrightarrow 4.5, 4 \leftrightarrow 6, 5 \leftrightarrow 7.5$
A positive enlargement preserves the ordering of shortest, middle and longest sides.
2
Compare all corresponding ratios.
$\begin{aligned} \frac{4.5}{3} & = 1.5 \\ \frac{6}{4} & = 1.5 \\ \frac{7.5}{5} & = 1.5 \end{aligned}$
Proportional SSS requires agreement of all three side ratios.
3
State the test and factor.
$k = 1.5 from the first triangle to the second$
All three ratios agree, so the triangles are similar by proportional SSS.

Answer

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

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

Yes, by AA.

The $30^{\circ}$ and $80^{\circ}$ angles provide two matching pairs, so AA gives similarity. In both triangles the remaining angle is $180^{\circ} - 30^{\circ} - 80^{\circ} = 70^{\circ}$ .

Keep practising

6 worksheet questions · 15 marks

Open worksheet 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 $△ A B C \sim △ D E F$ , $A B = 5$ cm, $D E = 15$ cm and $B C = 7$ cm. Find $E F$ .
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
$\frac{8}{4} = \frac{10}{5} = \frac{12}{6} = 2$ .
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 $\frac{11}{5} = 2.2$ .
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
$\frac{10}{6} = \frac{15}{9} = \frac{5}{3}$ , and the included angles match.
Yes, by proportional SAS.
Given $△ A B C \sim △ D E F$ , $A B = 5$ cm, $D E = 15$ cm and $B C = 7$ cm. Find $E F$ .
3 marks
Scale factor is $\frac{15}{5} = 3$ .
$E F = 7 (3) = 21$ 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 ↗

In this lesson

Understand the idea.

Three ways to establish the same shape

State a correspondence first

See the method in action.

Angles lead to a length

Establish the triangle correspondence.

Calculate the factor from ABC to DEF.

Scale the side corresponding to AC.

Check all three ratios

Pair sides by their relative lengths.

Compare all corresponding ratios.

State the test and factor.

Try it yourself.

Check your understanding

Keep practising