Similar Triangles — AA, SAS & SSS Similarity

Similar triangles are triangles with the same shape but not necessarily the same size. In a similar pair, corresponding angles are equal and corresponding sides are proportional.

The main reason this matters is practical: once you prove two triangles are similar, one scale factor can unlock every matching side. That is why similar triangles show up in geometry proofs, scale drawings, and shadow problems.

What similar triangles mean

If $\triangle ABC$ is similar to $\triangle DEF$, then the order matters. It tells you that angle $A$ matches angle $D$, angle $B$ matches angle $E$, and angle $C$ matches angle $F$.

From that matching, the corresponding sides satisfy

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

These ratios all describe the same scale factor. If the scale factor from $\triangle ABC$ to $\triangle DEF$ is $2$, then each side in $\triangle DEF$ is twice the matching side in $\triangle ABC$.

How AA, SAS, and SSS similarity work

AA similarity

If two angles of one triangle match two angles of another triangle, the triangles are similar.

This works because the third angles must also match, since the angles in any triangle add to $180^\circ$.

SAS similarity

If two pairs of corresponding sides are proportional and the included angle between those sides is equal, the triangles are similar.

The word "included" matters. The equal angle must sit between the two sides you are comparing. If the equal angle is somewhere else, SAS does not apply.

SSS similarity

If all three pairs of corresponding sides are proportional, the triangles are similar.

This is often the cleanest test when no angles are given, but only if the side pairs are matched correctly.

Worked example: use SAS to find a missing side

Suppose two triangles have an included angle of $40^\circ$. In the smaller triangle, the sides around that angle are $6$ and $12$. In the larger triangle, the matching sides are $10$ and $20$.

First check the side ratios:

$$\frac{6}{10} = \frac{12}{20} = \frac{3}{5}$$

Since the included angle is also equal, the triangles are similar by SAS.

Now suppose the third side of the smaller triangle is $9$, and the corresponding third side of the larger triangle is $x$. Use the scale factor from small to large:

$$\frac{10}{6} = \frac{20}{12} = \frac{5}{3}$$

So

$$x = 9 \cdot \frac{5}{3} = 15$$

The missing side is $15$. The key idea is not the arithmetic. It is the setup: prove similarity first, then use one consistent scale factor on corresponding sides.

Common mistakes when proving triangles are similar

Mixing up similar and congruent

Similar triangles have the same shape. Congruent triangles have the same shape and the same size. Congruent triangles are a special case of similar triangles with scale factor $1$.

Using the wrong side pairs

A correct proportion uses corresponding sides only. If the vertex order is wrong, the algebra can look neat while the setup is still wrong.

Flipping one ratio but not the others

If you write one ratio as small to large, the other ratios must also be small to large. Mixing directions inside the same equation creates wrong answers even when the triangles really are similar.

Treating SSA as a similarity test

AA, SAS, and SSS are valid similarity tests. SSA is not enough by itself in general, because the same side data can fit more than one triangle.

Forgetting that area scales differently

If side lengths scale by a factor of $k$, then areas scale by a factor of $k^2$. A triangle that is twice as wide is not just twice the area.

Where similar triangles are used

Similar triangles show up in geometry, maps, scale drawings, shadows, surveying, and coordinate geometry. They are especially useful when one triangle is easier to measure than another but the shapes match.

You also see them inside bigger proofs. The Pythagorean theorem, right-triangle altitude relationships, and some trigonometry ideas all become easier once similarity is recognized.

Try a similar problem

Try your own version with side pairs $8$ and $12$ in one triangle and $14$ and $21$ in another, with the included angle equal in both triangles. First prove similarity, then find the matching third side if the smaller one is $10$.

If you want one natural next step, try solving a similar problem where you must decide first whether the information fits AA, SAS, or SSS before setting up any proportion.