Why is it AA and not AAA?

Because the third pair of angles gives no new information. Once two pairs match, the Triangle Sum Theorem forces the third pair to match as well, since all three angles in each triangle add to 180^. Writing AAA is not wrong mathematically, but AA is the standard name because two angle pairs are all you ever need to state.

What is the difference between SAS congruence and SAS similarity?

SAS congruence requires the two pairs of sides to be equal in length and the included angles congruent, producing identical triangles. SAS similarity only requires the two pairs of sides to be in the same ratio, with the included angles congruent, producing triangles of the same shape but possibly different sizes. Congruence is the special case where the scale factor equals 1.

Why isn't there an SSA similarity criterion?

Two proportional sides plus a congruent angle that is not between them can produce two genuinely different triangles — the third side can swing to two different positions. This is the same ambiguity that makes SSA fail for congruence. Always confirm the given angle sits between the two sides whose ratio you used.

How do I know which sides correspond when the problem gives no diagram?

If a similarity statement like ABC DEF is given, read the letters in order: AB pairs with DE, BC with EF, and AC with DF. If no statement is given, list each triangle's sides from shortest to longest and match them in that order, since a dilation scales every length by the same factor and cannot reorder them.

GEOM-6.3

Triangle Similarity: AA, SSS & SAS

Learn how AA, SSS and SAS similarity prove two triangles are similar in Geometry 6.3, then use the scale factor to solve for missing side lengths.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Triangle Similarity: AA, SSS & SAS, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Triangle Similarity: AA, SSS & SAS free

What this lesson covers

You already know that similar polygons have congruent corresponding angles and proportional corresponding sides. For triangles, that is six separate facts to check — three angle pairs and three side ratios. The good news: triangles are rigid enough that you never have to check all six. Three shortcuts, called AA, SSS and SAS similarity, each let you establish similarity from just two or three pieces of information.

This lesson shows you exactly what each criterion requires, how to write a correct similarity statement, and how to turn that statement into a proportion that solves for an unknown side. These skills come back constantly — in the side-splitter theorem, in right-triangle trigonometry, and in any real problem that uses shadows, scale drawings or indirect measurement.

What Similarity Requires, and Why Shortcuts Exist

Two triangles are similar when their corresponding angles are congruent and their corresponding sides are proportional. Writing

\triangle ABC \sim \triangle DEF

is a compact way of stating six facts at once:

\angle A \cong \angle D,\quad \angle B \cong \angle E,\quad \angle C \cong \angle F,\quad \frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}

That common ratio is the scale factor, usually called

k

. If

k=1

the triangles are also congruent, so congruence is just the special case of similarity where nothing is enlarged or shrunk.

Triangles are special because their shape is completely determined by very little information. Fix all three angles and you have fixed the shape — only the size is left free. Fix the ratios of all three sides and the angles have no choice but to match. This rigidity is why three shortcuts work for triangles but not for quadrilaterals: a square and a non-square rhombus have all four side ratios equal to 1 and are still not similar.

The order of the letters in a similarity statement carries meaning.

\triangle ABC \sim \triangle DEF

says

A

corresponds to

D

B

E

, and

C

F

. Writing

\triangle ABC \sim \triangle EFD

instead makes a completely different claim. A large share of wrong answers in this unit trace back to a correspondence that was written carelessly, not to bad arithmetic.

AA Similarity: Two Angles Are Enough

The AA (Angle-Angle) criterion says that if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. You need no side information at all.

Why two and not three? Because of the Triangle Sum Theorem. If

\angle A \cong \angle D

and

\angle B \cong \angle E

, then

m\angle C = 180^\circ - m\angle A - m\angle B = 180^\circ - m\angle D - m\angle E = m\angle F

so the third pair matches automatically. That is why textbooks say AA rather than AAA — the third angle is free information.

AA is by far the most-used criterion because congruent angles are easy to spot in a diagram even when no lengths are given:

Where the angle pair comes from	What you cite
Two lines crossing at a point	Vertical angles are congruent
A transversal cutting parallel lines	Alternate interior or corresponding angles
Two triangles sharing a vertex angle	Reflexive property
Both triangles have a right angle	All right angles are congruent

A classic setup is a figure where

\overline{DE} \parallel \overline{BC}

inside

\triangle ABC

. The shared angle at

A

plus one pair of corresponding angles gives AA immediately.

One caution: AA proves similarity, never congruence. Two triangles can have identical angles and wildly different sizes. Students sometimes finish an AA argument and then write that a pair of sides is congruent — that step is not justified. AA gives you proportional sides, and you still need one known pair of corresponding lengths to pin down the scale factor.

SSS and SAS Similarity: Working With Ratios

When a diagram gives lengths instead of angle marks, use one of the two ratio-based criteria.

SSS Similarity: if all three pairs of corresponding sides are proportional, the triangles are similar. Check that the three ratios simplify to the same number.

SAS Similarity: if two pairs of corresponding sides are proportional and the included angles — the angles formed between those two sides — are congruent, the triangles are similar.

Criterion	What you must have	Easy mistake
AA	Two pairs of congruent angles	Claiming congruent sides afterward
SSS	Three equal side ratios	Pairing sides in the wrong order
SAS	Two equal side ratios plus congruent included angles	Using a non-included angle

To pair sides correctly when no correspondence is given, order each triangle's sides from shortest to longest and match them in that order. The smallest side of one triangle must correspond to the smallest side of the other, since a dilation scales every length by the same factor.

There is no SSA similarity criterion, for exactly the reason there is no SSA congruence criterion: two proportional sides and a congruent angle outside them can produce two different triangles. So in SAS work, always confirm the angle sits between the two sides you used.

Finally, note that these criteria are named to echo the congruence theorems, but they mean something different. SAS congruence needs the two sides to be equal in length; SAS similarity needs them only to be in the same ratio. Some teachers write AA~, SSS~ and SAS~ with a tilde to keep the two families apart.

Using the Scale Factor to Find Missing Lengths

Once similarity is established, every pair of corresponding sides shares the same ratio, and that is your solving tool.

Step one: write the similarity statement in correct correspondence order. Step two: compute the scale factor from a pair of sides whose lengths you know both of. Step three: set up a proportion that keeps the two triangles in consistent positions — either both first triangles on top, or both second triangles on top. Step four: cross multiply and solve.

For

\triangle ABC \sim \triangle DEF

with scale factor

k = \frac{AB}{DE}

, every unknown follows from

\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k

A reliable habit is to write the proportion in words first, such as

\frac{\text{short side of big}}{\text{short side of small}} = \frac{\text{long side of big}}{\text{long side of small}}

, then substitute numbers. This prevents the most common error: flipping one fraction and getting a value that is too small when it should be larger, or vice versa. Always sanity-check the answer. If

k > 1

the first triangle is the bigger one, so every side you find for it should exceed its partner.

Two extensions worth remembering: perimeters of similar triangles are in the ratio

k

, while areas are in the ratio

k^2

. So doubling every side multiplies the area by four.

Overlapping triangles cause trouble here. When one triangle sits inside another and they share part of a side, the shared segment is a part of the larger triangle's side, not the whole side. Redraw the two triangles separately before writing any ratio.

Where Students Actually Go Wrong

Similarity problems are rarely lost on arithmetic. Here are the failure points to watch.

Assuming similarity from how the picture looks. Diagrams are not drawn to scale unless the problem says so. You must name a criterion and cite the specific angles or ratios that satisfy it.

Using a non-included angle in SAS. If the given angle touches only one of the two sides in your ratio pair, SAS does not apply. Look for a third side length instead and try SSS, or find a second angle and use AA.

Mismatched correspondence in SSS. Suppose one triangle has sides 6, 8, 10 and another has 9, 12, 15. Comparing

\frac{6}{12}

\frac{8}{9}

and

\frac{10}{15}

gives three different values and a false conclusion of "not similar." Ordered correctly,

\frac{6}{9}=\frac{8}{12}=\frac{10}{15}=\frac{2}{3}

, and the triangles are similar.

Adding instead of scaling. Going from sides 6, 8, 10 to 9, 12, 15 is not "add 3" — adding 3 to each would give 9, 11, 13, which is a different shape. Similarity is multiplicative.

Stopping at similarity when the question asks for a length, or reporting the scale factor when the question asks for a side. Read the final line of the problem again before you write your answer.

Confusing similarity with congruence in a proof. After AA, you may conclude corresponding sides are proportional. Only after showing

k = 1

, or using a congruence theorem, may you say sides are equal.

Key terms

Similar triangles.: Two triangles whose corresponding angles are congruent and whose corresponding sides are proportional; written with the symbol $\sim$ .
Scale factor.: The constant ratio $k$ between the lengths of corresponding sides of two similar figures, computed as a length in the first figure divided by the matching length in the second.
AA Similarity Criterion.: If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.
SSS Similarity Criterion.: If all three pairs of corresponding sides of two triangles are proportional, the triangles are similar.
SAS Similarity Criterion.: If two pairs of corresponding sides are proportional and the included angles are congruent, the triangles are similar.
Included angle.: The angle formed at the vertex where two named sides of a triangle meet; required to be congruent in SAS similarity.
Correspondence.: The pairing of vertices between two triangles, encoded by the order of letters in a similarity statement such as $\triangle ABC \sim \triangle DEF$ .
Proportion.: An equation stating that two ratios are equal, such as $\frac{AB}{DE}=\frac{BC}{EF}$ , solved by cross multiplication.

Worked example

\triangle ABC

AB = 6

AC = 9

, and

m\angle A = 52^\circ

. In

\triangle DEF

DE = 4

DF = 6

, and

m\angle D = 52^\circ

. Show that the triangles are similar, state the scale factor, and find

BC

EF = 5

Start by checking which sides are given. In

\triangle ABC

the two known sides are

AB

and

AC

, which meet at vertex

A

. In

\triangle DEF

the two known sides are

DE

and

DF

, which meet at vertex

D

. The given angles are exactly the angles at

A

and

D

, so these are the included angles. That points to SAS similarity.

Now test the two ratios, matching

AB

with

DE

and

AC

with

DF

\frac{AB}{DE} = \frac{6}{4} = \frac{3}{2} \qquad \frac{AC}{DF} = \frac{9}{6} = \frac{3}{2}

The ratios are equal, and the included angles are congruent because both measure

52^\circ

. By SAS similarity,

\triangle ABC \sim \triangle DEF

.

The scale factor from

\triangle DEF

\triangle ABC

k = \frac{3}{2} = 1.5

, so

\triangle ABC

is the larger triangle.

To find

BC

, use the correspondence:

B

pairs with

E

and

C

pairs with

F

, so

\overline{BC}

corresponds to

\overline{EF}

. Set up the proportion with

\triangle ABC

on top in both fractions:

\frac{BC}{EF} = \frac{AB}{DE} \quad\Rightarrow\quad \frac{BC}{5} = \frac{3}{2}

Cross multiply:

2 \cdot BC = 15

, so

BC = 7.5

.

Check it for reasonableness. Since

k = 1.5 > 1

, every side of

\triangle ABC

should be longer than its partner in

\triangle DEF

, and

7.5 > 5

. The answer is consistent.

Practice questions

\triangle PQR

PQ = 8

QR = 12

, and

PR = 16

. In

\triangle STU

ST = 6

TU = 9

, and

SU = 12

. Which statement is correct?

The triangles are not similar, because $8 - 6 \neq 16 - 12$ .
$\triangle PQR \sim \triangle STU$ by SSS similarity, with scale factor $\frac{4}{3}$ .
$\triangle PQR \sim \triangle STU$ by SAS similarity, with scale factor $\frac{3}{4}$ .
$\triangle PQR \sim \triangle SUT$ by SSS similarity, with scale factor $\frac{4}{3}$ .

Answer: $\triangle PQR \sim \triangle STU$ by SSS similarity, with scale factor $\frac{4}{3}$ .

Compare the three ratios in matching order:

\frac{PQ}{ST} = \frac{8}{6} = \frac{4}{3}

\frac{QR}{TU} = \frac{12}{9} = \frac{4}{3}

, and

\frac{PR}{SU} = \frac{16}{12} = \frac{4}{3}

. All three agree, so SSS similarity applies with

k = \frac{4}{3}

. The first option uses subtraction, but similarity is multiplicative, not additive. The third names SAS even though no angle measure was given. The fourth scrambles the correspondence:

\overline{PR}

must pair with

\overline{SU}

, not with

\overline{ST}

\triangle ABC

m\angle A = 90^\circ

and

m\angle B = 35^\circ

. In

\triangle DEF

m\angle D = 90^\circ

and

m\angle F = 55^\circ

. Explain why the triangles are similar, write the similarity statement, and then find

DF

given that

AB = 12

AC = 5

, and

DE = 18

Answer: $\triangle ABC \sim \triangle DEF$ by AA, and $DF = 7.5$ .

First find the missing angles. In

\triangle ABC

m\angle C = 180 - 90 - 35 = 55^\circ

. In

\triangle DEF

m\angle E = 180 - 90 - 55 = 35^\circ

. Now match:

\angle A \cong \angle D

(both

90^\circ

) and

\angle B \cong \angle E

(both

35^\circ

), so by AA the triangles are similar and the correspondence is

\triangle ABC \sim \triangle DEF

. Note that you must match by angle measure, not by letter position alone — here it happens to work out, but if

\angle F

had been

35^\circ

the correspondence would have been

\triangle ABC \sim \triangle DFE

. Next, the scale factor from

\triangle ABC

\triangle DEF

comes from the known pair

AB

and

DE

\frac{DE}{AB} = \frac{18}{12} = \frac{3}{2}

. Since

\overline{AC}

corresponds to

\overline{DF}

, solve

\frac{DF}{5} = \frac{3}{2}

, giving

DF = 7.5

. Because

\triangle DEF

is the larger triangle,

DF

should exceed

AC = 5

, and it does.

\triangle GHI

and

\triangle JKL

, you know that

\frac{GH}{JK} = \frac{HI}{KL} = \frac{5}{2}

. Which additional fact would prove the triangles similar by SAS?

$\angle G \cong \angle J$
$\angle H \cong \angle K$
$\angle I \cong \angle L$
$\frac{GI}{JL} = \frac{5}{2}$

Answer: $\angle H \cong \angle K$

SAS similarity requires the congruent angles to be the ones included between the two proportional side pairs. The sides

\overline{GH}

and

\overline{HI}

both contain vertex

H

, so

\angle H

is the included angle in the first triangle; likewise

\overline{JK}

and

\overline{KL}

meet at

K

. Angles

G

J

I

and

L

are non-included, and two sides plus a non-included angle is not a valid similarity criterion. The last option is true information and would in fact prove similarity — but by SSS, not SAS, so it does not answer the question asked.

FAQ

Why is it AA and not AAA?: Because the third pair of angles gives no new information. Once two pairs match, the Triangle Sum Theorem forces the third pair to match as well, since all three angles in each triangle add to $180^\circ$ . Writing AAA is not wrong mathematically, but AA is the standard name because two angle pairs are all you ever need to state.
What is the difference between SAS congruence and SAS similarity?: SAS congruence requires the two pairs of sides to be equal in length and the included angles congruent, producing identical triangles. SAS similarity only requires the two pairs of sides to be in the same ratio, with the included angles congruent, producing triangles of the same shape but possibly different sizes. Congruence is the special case where the scale factor equals 1.
Why isn't there an SSA similarity criterion?: Two proportional sides plus a congruent angle that is not between them can produce two genuinely different triangles — the third side can swing to two different positions. This is the same ambiguity that makes SSA fail for congruence. Always confirm the given angle sits between the two sides whose ratio you used.
How do I know which sides correspond when the problem gives no diagram?: If a similarity statement like $\triangle ABC \sim \triangle DEF$ is given, read the letters in order: $\overline{AB}$ pairs with $\overline{DE}$ , $\overline{BC}$ with $\overline{EF}$ , and $\overline{AC}$ with $\overline{DF}$ . If no statement is given, list each triangle's sides from shortest to longest and match them in that order, since a dilation scales every length by the same factor and cannot reorder them.

Learn this with a teacher, not a page

The Crimsora tutor teaches Triangle Similarity: AA, SSS & SAS live — explaining on a whiteboard, asking you questions, and adapting to where you get stuck.

Try a live session free