Math Core

AA similarity is perfect when you know angles. But sometimes all you know are side lengths, or two sides and the angle between them. Two more shortcuts, SSS and SAS similarity, handle those cases. They mirror the SSS and SAS congruence shortcuts, with "congruent sides" replaced by "proportional sides."

Side-Side-Side (SSS) Similarity

SSS Similarity

If the three sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar.

In symbols: if $\dfrac{AB}{DE} = \dfrac{BC}{EF} = \dfrac{AC}{DF}$ , then $\triangle ABC \sim \triangle DEF$ .

Why does it work? Dilate $\triangle ABC$ by the common ratio $k = \dfrac{DE}{AB}$ . The image has sides $k \cdot AB = DE$ , $k \cdot BC = EF$ and $k \cdot AC = DF$ , so it is congruent to $\triangle DEF$ by SSS congruence. A dilation followed by a rigid motion takes one triangle onto the other.

To test SSS, sort each triangle's sides from shortest to longest and compare in that order: shortest to shortest, middle to middle, longest to longest. If all three ratios are equal, the triangles are similar. If even one ratio is different, they're not.

Worked example: Testing with SSS

a. Is a triangle with sides $6$ , $8$ , $10$ similar to a triangle with sides $9$ , $12$ , $15$ ?

$\frac{9}{6} = 1.5, \qquad \frac{12}{8} = 1.5, \qquad \frac{15}{10} = 1.5.$

All three ratios are equal, so the triangles are similar by SSS, with scale factor $1.5$ .

b. Is a triangle with sides $6$ , $8$ , $10$ similar to a triangle with sides $9$ , $12$ , $16$ ?

$\dfrac{9}{6} = 1.5$ and $\dfrac{12}{8} = 1.5$ , but $\dfrac{16}{10} = 1.6$ . One ratio is off, so the triangles are not similar.

Worked example: Sorting first and naming the triangles

$\triangle ABC$ has $AB = 12$ , $BC = 8$ , $CA = 10$ . $\triangle PQR$ has $PQ = 15$ , $QR = 18$ , $RP = 12$ . Are they similar? If so, write a similarity statement.

Sort the sides:

	shortest	middle	longest
$\triangle ABC$	$BC = 8$	$CA = 10$	$AB = 12$
$\triangle PQR$	$RP = 12$	$PQ = 15$	$QR = 18$
ratio	$1.5$	$1.5$	$1.5$

The triangles are similar by SSS. To name them, match the vertex opposite each pair of sides. The vertex opposite the shortest side is $A$ in one triangle and $Q$ in the other; opposite the middle side, $B$ and $R$ ; opposite the longest side, $C$ and $P$ . So

$\triangle ABC \sim \triangle QRP.$

Check one pair: $\overline{AB}$ should match $\overline{QR}$ , and indeed $\dfrac{18}{12} = 1.5$ .

Side-Angle-Side (SAS) Similarity

SAS Similarity

If an angle of one triangle is congruent to an angle of another triangle, and the sides including those angles are proportional, then the triangles are similar.

In symbols: if $\angle A \cong \angle D$ and $\dfrac{AB}{DE} = \dfrac{AC}{DF}$ , then $\triangle ABC \sim \triangle DEF$ . The angle must be the one between the two sides you compare.

Two sides and the included angle of each triangle.

Worked example: Testing with SAS

In the figure, $\angle A \cong \angle D$ (both $50^\circ$ ), $AB = 4$ , $AC = 6$ , $DE = 6$ and $DF = 9$ . Are the triangles similar? If $EF = 12$ , find $BC$ .

The congruent angles are included between the sides we know. Compare the ratios:

$\frac{AB}{DE} = \frac{4}{6} = \frac23, \qquad \frac{AC}{DF} = \frac{6}{9} = \frac23.$

They're equal, so $\triangle ABC \sim \triangle DEF$ by SAS. The scale factor from $\triangle DEF$ to $\triangle ABC$ is $\tfrac23$ , so $BC = \dfrac23 \cdot 12 = 8$ .

Proving lines parallel with similarity

SAS similarity often appears when one triangle sits inside another and they share an angle. Once the triangles are similar, their corresponding angles are congruent, which can prove that two lines are parallel.

Segment DE is parallel to side BC.

Worked example: A proof with a shared angle

Given: $D$ is on $\overline{AB}$ and $E$ is on $\overline{AC}$ , with $AD = 4$ , $AB = 10$ , $AE = 6$ , $AC = 15$ . Prove: $\overline{DE} \parallel \overline{BC}$ .

#	statement	reason
1	$AD = 4$ , $AB = 10$ , $AE = 6$ , $AC = 15$	Given
2	$\dfrac{AD}{AB} = \dfrac{4}{10} = \dfrac25$ and $\dfrac{AE}{AC} = \dfrac{6}{15} = \dfrac25$	Substitution and simplifying
3	$\dfrac{AD}{AB} = \dfrac{AE}{AC}$	Transitive Property of Equality
4	$\angle A \cong \angle A$	Reflexive Property of Congruence
5	$\triangle ADE \sim \triangle ABC$	SAS Similarity (steps 3, 4)
6	$\angle ADE \cong \angle ABC$	Corresponding angles of similar triangles are congruent
7	$\overline{DE} \parallel \overline{BC}$	Converse of the Corresponding Angles Postulate

Choosing a shortcut

Here are the three ways to prove triangles similar, next to their congruence cousins.

similarity shortcut	what you need	congruence version
AA	two pairs of congruent angles	ASA or AAS (which also need a side)
SSS	all three pairs of sides proportional	SSS (sides congruent)
SAS	one pair of congruent angles and the two sides around them proportional	SAS (sides congruent)

Common mistake

There is no SSA similarity. If the congruent angle is not between the two proportional sides, the triangles may or may not be similar. Always check that the angle is the one formed by the two sides you're comparing.

Tip

Cross-multiplying is the fastest way to compare two ratios. For $\dfrac{4}{6}$ and $\dfrac{6}{9}$ : $4 \cdot 9 = 36$ and $6 \cdot 6 = 36$ , so the ratios are equal.

Practice

Practice 1

A triangle has sides $5$ , $12$ , $13$ . Another has sides $15$ , $36$ , $39$ . Are the triangles similar?

Practice 2

A triangle has sides $4$ , $6$ , $8$ . Another has sides $6$ , $9$ , $14$ . Which is true?

Practice 3

A triangle has sides $3$ , $5$ and $6$ . A similar triangle has shortest side $4.5$ and longest side $9$ . What is the length of its middle side?