AA Similarity

Q: Why does AA work with only two angles?

Because the three angles of any triangle add to $180^\circ$. If two pairs of angles match, the third pair is forced to match too — so two pairs of angles carry exactly as much information as three.

Similar triangles have the same shape but not necessarily the same size. Formally, $\triangle ABC \sim \triangle DEF$ means all three pairs of corresponding angles are congruent, and all three pairs of corresponding sides are proportional — every side of the second triangle is the same multiple of its partner. That multiple is called the scale factor.

Similarity is one of the most useful ideas in geometry because it turns shapes into equations. If two triangles are similar, a proportion connects their sides — which is how you measure a tree from its shadow, and why sine and cosine work at all. The two skills you need: proving two triangles are similar, and setting up the proportion correctly once you know they are.

Three ways to prove triangles similar

AA (Angle-Angle): two pairs of corresponding angles are congruent. That's all it takes — since the angles of every triangle sum to $180^\circ$ , matching two pairs forces the third pair to match automatically. AA is by far the most common criterion on tests.

SSS similarity: all three pairs of corresponding sides have the same ratio. Compare shortest to shortest, middle to middle, longest to longest — if all three ratios are equal, the triangles are similar with that ratio as the scale factor.

SAS similarity: two pairs of sides have the same ratio, and the included angles — the angles between those sides — are congruent. Just like SAS congruence, the angle must sit between the two sides.

The two right triangles below are similar: $\dfrac{8}{4} = \dfrac{6}{3} = \dfrac{10}{5} = 2$ , so every side of the larger triangle is exactly twice its partner. Same shape, scale factor $2$ .

Setting up the proportion

The similarity statement tells you which parts correspond — by letter order. In $\triangle ABC \sim \triangle DEF$ , side $\overline{AB}$ pairs with $\overline{DE}$ , $\overline{BC}$ with $\overline{EF}$ , and $\overline{AC}$ with $\overline{DF}$ . Trust the letters, not the picture — diagrams are often rotated or flipped.

To find a missing side, write one ratio of known corresponding sides, set it equal to the ratio containing the unknown, and cross multiply. Keep the same triangle on top in both ratios: $\dfrac{AB}{DE} = \dfrac{BC}{EF}$ . Mixing which triangle goes in the numerator is the most common setup error.

Similar vs. congruent

Congruent triangles are a special case of similar triangles — same shape and same size, which means a scale factor of exactly $1$ . If two triangles are congruent they are automatically similar, but not the other way around.

This is also why AAA doesn't prove congruence: three matching angles lock in the shape but say nothing about size. Angles alone can only ever get you to similarity.

Worked examples

Example 1: proving similarity with AA

$\triangle ABC$ has angles measuring $58^\circ$ and $63^\circ$ . $\triangle DEF$ has angles measuring $63^\circ$ and $59^\circ$ . Are the triangles similar?

Find the third angle of

\triangle ABC

with the Triangle Angle Sum Theorem

180^\circ - 58^\circ - 63^\circ = 59^\circ

Now compare angle lists:

\triangle ABC

has

58^\circ, 63^\circ, 59^\circ

and

\triangle DEF

has

63^\circ

and

59^\circ

Two pairs of congruent angles (

63^\circ

and

59^\circ

) is enough

\triangle ABC \sim \triangle DEF \text{ by AA}

Answer: Yes — similar by AA

Example 2: checking SSS similarity

$\triangle ABC$ has sides $4$ , $6$ , and $8$ . $\triangle DEF$ has sides $6$ , $9$ , and $12$ . Are the triangles similar?

Compare shortest to shortest

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

Compare middle to middle

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

Compare longest to longest

\dfrac{8}{12} = \dfrac{2}{3}

All three ratios are equal, so the triangles are similar with scale factor

\dfrac{2}{3}

\triangle ABC \sim \triangle DEF \text{ by SSS similarity}

Answer: Yes — similar by SSS similarity, scale factor $\dfrac{2}{3}$

Example 3: solving for a missing side

$\triangle ABC \sim \triangle DEF$ with $AB = 6$ , $DE = 9$ , and $BC = 8$ . Find $EF$ .

Match corresponding sides by letter order and write the proportion

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

Substitute the known lengths

\dfrac{6}{9} = \dfrac{8}{EF}

Cross multiply

6 \cdot EF = 72

Divide both sides by

6

EF = 12

Answer: $EF = 12$

Example 4: indirect measurement with shadows

A $6$ -foot-tall person casts a $4$ -foot shadow. At the same moment, a tree casts an $18$ -foot shadow. How tall is the tree?

The sun's rays hit both at the same angle, so the two height-and-shadow triangles are similar by AA

Set up the proportion: height over shadow equals height over shadow

\dfrac{h}{18} = \dfrac{6}{4}

Cross multiply

4h = 108

Divide both sides by

4

h = 27

Answer: The tree is $27$ feet tall

Try one yourself

Common questions

What's the difference between similar and congruent triangles?

Congruent triangles match in both shape and size — corresponding sides are equal. Similar triangles match in shape only — corresponding angles are equal, and corresponding sides are proportional. Congruence is just similarity with a scale factor of $1$ .

Why does AA work with only two angles?

Because the three angles of any triangle add to $180^\circ$ . If two pairs of angles match, the third pair is forced to match too — so two pairs of angles carry exactly as much information as three.

How do I know which sides correspond?

Read the similarity statement's letter order: in $\triangle ABC \sim \triangle DEF$ , the first letters pair up, the second letters pair up, and so on, so $\overline{AB}$ corresponds to $\overline{DE}$ . If there's no statement, match each side by the angles at its endpoints — corresponding sides connect corresponding angles.

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