Triangle similarity

Warm-up

Three quick questions from earlier lessons.

Discover

Two triangles each have angles $40^\circ$ and $60^\circ$ . What is the third angle in each?

$80^\circ$ in both
It depends on the sizes

Key idea

Two angles are enough.

AA: two pairs of equal angles make triangles similar. (Also SSS similarity: all sides proportional; SAS similarity: two sides proportional and the included angles equal.) Then corresponding sides are proportional:

\dfrac{DE}{AB} = \dfrac{EF}{BC} = \dfrac{DF}{AC}

Worked example

$\triangle ABC \sim \triangle DEF$ , $AB = 4$ , $BC = 6$ , $DE = 10$ .

Scale factor

2.5

DE/AB.

EF = 15

EF = 2.5 × BC.

Your turn, guided

Your turn: $AB = 3$ , $AC = 5$ , $DE = 9$ . Find $DF$ .

DF = 15

5 × 3.

Practice ★

Two triangles each have a $76^\circ$ angle and a $50^\circ$ angle. What can you conclude?
1. They are similar (AA)
2. They are congruent
3. Nothing without side lengths

$\triangle ABC \sim \triangle DEF$ with $AB = 8$ , $BC = 7$ and $DE = 25$ . Find $EF$ .Not to scale
$\triangle ABC \sim \triangle DEF$ with $AB = 9$ , $BC = 3$ and $DE = 24$ . Find $EF$ .Not to scale

Watch out!

Practice ★★

Which figures are always similar to each other?
1. Any two squares (for example sides 7 and 14)
2. Any two rectangles
3. Any two isosceles triangles

Exit ticket

Three questions to check you have got it.

$\triangle ABC \sim \triangle DEF$ with $AB = 8$ , $BC = 7$ and $DE = 25$ . Find $EF$ .Not to scale
$\triangle ABC \sim \triangle DEF$ with $AB = 9$ , $BC = 3$ and $DE = 24$ . Find $EF$ .Not to scale
$\triangle ABC \sim \triangle DEF$ with $AB = 7$ , $BC = 9$ and $DE = 16$ . Find $EF$ .Not to scale

Summary card

Triangle similarity

AA (or SSS, SAS similarity) → corresponding sides proportional.

Watch out: AA works for triangles only.

End of the lesson

Next lesson: Proportional segments: the side-splitter theorem Back to the chapter

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