Similar Triangles and the AA Criterion

Same shape, different size

Two figures are similar when one is an enlargement of the other. Formally, one can be carried onto the other by a dilation followed by rigid motions.

Two things follow, and they are the two you actually use:

In similar figures	What holds
corresponding angles	equal
corresponding sides	all in the same ratio

That common ratio is the scale factor. Doubling every side doubles the figure; the angles never move.

$$\triangle ABC \sim \triangle DEF \;\Rightarrow\; \frac{DE}{AB} = \frac{EF}{BC} = \frac{DF}{AC}$$

The order of the letters carries the matching. Writing $\triangle ABC \sim \triangle DEF$ says $A$ pairs with $D$, $B$ with $E$, $C$ with $F$. Getting that order wrong is what produces upside-down proportions.

Why two angles are enough

To prove two general figures similar you would have to check every angle and every side ratio. For triangles, two angles settle it.

$$\angle A = \angle D \;\text{ and }\; \angle B = \angle E \;\Rightarrow\; \triangle ABC \sim \triangle DEF$$

The reason is the angle sum. The three angles of a triangle total $180°$, so once two pairs match, the third has no freedom left:

$$\angle C = 180° - \angle A - \angle B = 180° - \angle D - \angle E = \angle F$$

All three angles now match, which fixes the shape completely. The only thing still free is the size — and that is exactly what similarity allows.

This is a triangle-only privilege. Two rectangles both have four right angles, and a $2 \times 3$ rectangle is nothing like a $2 \times 30$ one. Matching angles constrain a triangle in a way they do not constrain anything with more sides.

Where the equal angles come from

In practice you rarely get told two angles are equal. You find them.

Situation	Why the angles match
a shared angle	the same angle is in both triangles
parallel lines cut by a line	corresponding angles are equal
vertical angles at a crossing	vertical angles are equal
two right angles	both are $90°$

A very common case is a line drawn parallel to one side of a triangle. It creates a small triangle inside the large one, sharing the apex angle, with the other angles equal because the lines are parallel.

So the two triangles are similar, and the parallel line cuts the two sides it crosses in the same ratio:

$$\frac{AD}{DB} = \frac{AE}{EC}$$

That result is the triangle proportionality theorem, and it is a corollary of AA rather than a separate fact to memorize.

Setting up the proportion

Once similarity is established, matching sides give an equation.

Triangle $ABC$ has $AB = 6$ and $BC = 9$. Triangle $DEF$ is similar with $DE = 8$. Find $EF$.

Match the sides by their position in the similarity statement:

$$\frac{DE}{AB} = \frac{EF}{BC} \quad\Rightarrow\quad \frac{8}{6} = \frac{EF}{9}$$ $$EF = 9 \times \frac{8}{6} = 12$$

Put corresponding sides in corresponding positions. The safest habit is to write the ratio as new over old on both sides, so both fractions are the scale factor.

Check the size: the scale factor is $\tfrac{8}{6}$, a bit over $1$, so every new side should be a bit longer than its partner. $12$ against $9$ fits.

Measuring what you cannot reach

This is the payoff, and it is why similarity is on every surveying and navigation course ever written.

A $6$ ft person casts a $4$ ft shadow. A flagpole casts a $30$ ft shadow at the same moment. How tall is the flagpole?

The sun’s rays arrive at the same angle for both, and both objects stand vertically. That is two equal angles, so the triangles are similar.

$$\frac{h}{30} = \frac{6}{4} \quad\Rightarrow\quad h = 30 \times 1.5 = 45 \text{ ft}$$

The measurement you cannot take is replaced by three you can. Nothing here needs a ladder.

The same argument works with a mirror on the ground, with a sighting stick held at arm’s length, and with the shadow of a mountain. All of it is AA.

Worked examples

Common mistakes

Practice problems