SSS & SAS Similarity

AA similarity works when you know angles, but plenty of problems hand you side lengths instead. That is what SSS and SAS similarity are for. SSS similarity checks whether all three side ratios are equal. SAS similarity checks two side ratios and the angle included between those sides.

These are the similarity versions of the congruence shortcuts you already know — with one key difference. Congruence asks whether corresponding sides are equal. Similarity asks whether corresponding sides are proportional, all scaled by the same factor.

SSS similarity: three ratios, all equal

Line up the sides before you divide: shortest with shortest, middle with middle, longest with longest. Then compute all three ratios. If every ratio equals the same number, the triangles are similar by SSS, and that number is the scale factor.

All three ratios must match. Two matching ratios and one that disagrees means the triangles are not similar — stopping after two checks is the most common way to get these wrong.

SAS similarity: two ratios and the included angle

SAS similarity needs a pair of congruent angles plus proportional sides on both arms of that angle. In symbols: if $\angle M \cong \angle R$ and $\dfrac{MN}{RS} = \dfrac{MP}{RT}$ , then the triangles are similar by SAS.

The angle must be included — sandwiched between the two sides you compared. An angle hanging off to the side of the proportional pairs proves nothing.

A useful special case: two triangles that share an angle. If $D$ sits on $\overline{AB}$ and $E$ sits on $\overline{AC}$ , then $\triangle ADE$ and $\triangle ABC$ share $\angle A$ automatically, and you only need to check that the sides along each ray are proportional.

The figure shows that shared-angle case: $\triangle ADE$ sits inside $\triangle ABC$ , and both use $\angle A$ .

Choosing the right shortcut

Count what you are given. Three pairs of sides — test SSS. Two pairs of sides plus the angle between them — test SAS. Two pairs of angles — that is AA, no sides needed.

If a check fails, the shortcut does not apply, and often that is the answer itself: the triangles are not similar. Show the ratio that breaks and you have a complete justification.

Worked examples

Example 1: SSS similarity

$\triangle ABC$ has sides $5$ , $6$ , and $10$ . $\triangle DEF$ has sides $10$ , $12$ , and $20$ . Are the triangles similar?

Compare the shortest sides

\dfrac{5}{10} = \dfrac{1}{2}

Compare the middle sides

\dfrac{6}{12} = \dfrac{1}{2}

Compare the longest sides

\dfrac{10}{20} = \dfrac{1}{2}

All three ratios equal

\dfrac{1}{2}

, so the triangles are similar by SSS

Answer: Yes — $\triangle ABC \sim \triangle DEF$ by SSS similarity, with scale factor $\dfrac{1}{2}$

Example 2: SAS similarity

In $\triangle MNP$ and $\triangle RST$ , $\angle M \cong \angle R$ , $MN = 4$ , $MP = 6$ , $RS = 6$ , and $RT = 9$ . Are the triangles similar?

Compare the sides on one arm of the angle

\dfrac{MN}{RS} = \dfrac{4}{6} = \dfrac{2}{3}

Compare the sides on the other arm

\dfrac{MP}{RT} = \dfrac{6}{9} = \dfrac{2}{3}

The included angles are congruent and both ratios match

\angle M \cong \angle R

Answer: Yes — $\triangle MNP \sim \triangle RST$ by SAS similarity

Example 3: when the ratios do not match

$\triangle PQR$ has sides $3$ , $4$ , and $6$ . $\triangle XYZ$ has sides $6$ , $8$ , and $10$ . Are the triangles similar?

Compare the shortest sides

\dfrac{6}{3} = 2

Compare the middle sides

\dfrac{8}{4} = 2

Compare the longest sides

\dfrac{10}{6} = \dfrac{5}{3}

The third ratio breaks the pattern

2 \neq \dfrac{5}{3}

Answer: No — the triangles are not similar, because SSS requires all three ratios to be equal

Try one yourself

Common questions

Do I have to compare the sides in size order?

Yes — that is the safe way. Match shortest to shortest, middle to middle, longest to longest. If you pair sides randomly, you can get unequal ratios from triangles that really are similar, or accidentally match ratios that do not correspond.

What is the difference between SAS congruence and SAS similarity?

SAS congruence needs the two pairs of sides to be equal. SAS similarity only needs them to be proportional — both pairs scaled by the same factor. Both versions require the congruent angle to be included between those sides.

Is two equal ratios enough for SSS?

No. SSS similarity is a three-ratio check. Triangles with sides $3, 4, 5$ and $6, 8, 9$ pass two checks ( $2$ and $2$ ) but fail the third ( $\dfrac{9}{5} \neq 2$ ), so they are not similar.

Is there an SSA or AAA similarity shortcut?

AAA is really just AA — once two pairs of angles are congruent, the third pair is automatic. SSA is not a valid shortcut for similarity or congruence: the angle has to be included between the two sides for SAS to work.

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SSS & SAS Similarity

SSS similarity: three ratios, all equal

SAS similarity: two ratios and the included angle

Choosing the right shortcut

Worked examples

Example 1: SSS similarity

Example 2: SAS similarity

Example 3: when the ratios do not match

Try one yourself

Common questions

Keep learning

Similar Polygons & Scale Factor

The Triangle Proportionality (Side-Splitter) Theorem

Similarity Proofs & Indirect Measurement

Parts of Similar Triangles