Math Core

Checking that two polygons are similar normally takes a lot of work: every angle and every side ratio. For triangles there's a huge shortcut. Two pairs of matching angles are enough, and that one fact powers most of the similarity work you'll do in geometry.

Why two angles are enough

Suppose $\angle A \cong \angle D$ and $\angle B \cong \angle E$ in triangles $ABC$ and $DEF$ . The angles of any triangle add to $180^\circ$ , so

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

The third pair of angles matches automatically. But what about the sides? Here's where dilations come in. Dilate $\triangle ABC$ by the scale factor $k = \dfrac{DE}{AB}$ . The image has the same angles as $\triangle ABC$ , and its side matching $\overline{AB}$ now has length $DE$ . So the image and $\triangle DEF$ have two congruent angles and a congruent side between them, which makes them congruent by ASA. A dilation followed by a rigid motion maps $\triangle ABC$ onto $\triangle DEF$ , and that's the definition of similar.

Angle-Angle (AA) Similarity

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Notice what AA does not need: no side lengths at all. Once you know the triangles are similar, you get all the side proportions for free.

Common mistake

AA works for triangles only. Two quadrilaterals can have all four angles equal without being similar: every rectangle has four right angles, but a $1 \times 1$ square and a $1 \times 4$ rectangle are different shapes.

Writing the correspondence

Similar triangles are only useful if you match the vertices correctly. Pair up vertices whose angles are congruent, then write the similarity statement in that order.

Two angles of triangle ABC and two angles of triangle DEF are marked.

Worked example: Deciding with angle measures

In $\triangle ABC$ , $m\angle A = 50^\circ$ and $m\angle B = 70^\circ$ . In $\triangle DEF$ , $m\angle D = 70^\circ$ and $m\angle E = 60^\circ$ . Are the triangles similar? If so, write a similarity statement.

Find the missing angles. $m\angle C = 180^\circ - 50^\circ - 70^\circ = 60^\circ$ and $m\angle F = 180^\circ - 70^\circ - 60^\circ = 50^\circ$ .

Now match equal angles: $A$ and $F$ are $50^\circ$ , $B$ and $D$ are $70^\circ$ , $C$ and $E$ are $60^\circ$ . The triangles are similar by AA:

$\triangle ABC \sim \triangle FDE.$

Writing $\triangle ABC \sim \triangle DEF$ would be wrong, because $\angle A$ is not congruent to $\angle D$ .

Where the matching angles come from

In real problems, nobody hands you angle measures. You find congruent angles using facts you already know:

Parallel lines: corresponding angles and alternate interior angles are congruent.
Vertical angles are congruent.
A shared angle: when two triangles overlap, they may have an angle in common (Reflexive Property).
Right angles are all congruent.

The most common setup is a segment drawn parallel to one side of a triangle.

Segment DE is parallel to side BC.

Worked example: A proof with a parallel segment

Given: $\overline{DE} \parallel \overline{BC}$ , with $D$ on $\overline{AB}$ and $E$ on $\overline{AC}$ . Prove: $\triangle ADE \sim \triangle ABC$ .

#	statement	reason
1	$\overline{DE} \parallel \overline{BC}$	Given
2	$\angle ADE \cong \angle ABC$	Corresponding Angles Postulate
3	$\angle A \cong \angle A$	Reflexive Property of Congruence
4	$\triangle ADE \sim \triangle ABC$	AA Similarity (steps 2, 3)

Now suppose $AD = 4$ , $AB = 10$ and $DE = 6$ . The triangles are similar with $D \leftrightarrow B$ and $E \leftrightarrow C$ , so

$\frac{DE}{BC} = \frac{AD}{AB} \implies \frac{6}{BC} = \frac{4}{10} \implies 4 \cdot BC = 60 \implies BC = 15.$

Careful: the side of the big triangle is the whole side $AB$ , not just the piece $DB$ .

A second common setup is the "hourglass" (or "bowtie"): two segments cross between two parallel lines.

AB is parallel to DE. Segments AE and BD cross at C.

Worked example: An hourglass

In the figure, $\overline{AB} \parallel \overline{DE}$ , and $\overline{AE}$ and $\overline{BD}$ meet at $C$ . If $AC = 6$ , $CE = 9$ and $AB = 8$ , find $DE$ .

$\angle ACB \cong \angle ECD$ because they are vertical angles. $\angle A \cong \angle E$ because they are alternate interior angles for the parallel lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{DE}$ cut by $\overleftrightarrow{AE}$ . By AA, $\triangle ACB \sim \triangle ECD$ , with $A \leftrightarrow E$ , $C \leftrightarrow C$ , $B \leftrightarrow D$ . Then

$\frac{DE}{AB} = \frac{CE}{AC} = \frac{9}{6} \implies DE = \frac{9}{6} \cdot 8 = 12.$

Indirect measurement

On a sunny day, the sun's rays hit the ground at the same angle everywhere nearby. A vertical person and a vertical tree both make right angles with flat ground. So the triangle formed by the person and their shadow is similar to the triangle formed by the tree and its shadow, by AA.

Worked example: How tall is the tree?

A person $1.6$ m tall casts a shadow $2$ m long. At the same moment, a tree casts a shadow $15$ m long. How tall is the tree?

Heights match heights and shadows match shadows:

$\frac{h}{1.6} = \frac{15}{2} \implies h = 1.6 \cdot 7.5 = 12 \text{ m}.$

Tip

Before writing a proportion, write the similarity statement and read the matching sides off it. If you write $\dfrac{\text{small}}{\text{big}}$ on the left, write $\dfrac{\text{small}}{\text{big}}$ on the right too.

Practice

Practice 1

Which pair of triangles is not necessarily similar?

Practice 2

In $\triangle ABC$ , $m\angle A = 35^\circ$ and $m\angle B = 85^\circ$ . In $\triangle XYZ$ , $m\angle X = 35^\circ$ and $m\angle Z = 60^\circ$ . Which statement is correct?

Practice 3

In the proof that $\triangle ADE \sim \triangle ABC$ when $\overline{DE} \parallel \overline{BC}$ , which reason justifies $\angle AED \cong \angle ACB$ ?

Practice 4

In $\triangle ABC$ , $\overline{DE} \parallel \overline{BC}$ with $D$ on $\overline{AB}$ and $E$ on $\overline{AC}$ . If $AD = 3$ , $DB = 6$ and $DE = 5$ , find $BC$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

A person $5$ feet tall casts a $3$ -foot shadow. At the same time, a flagpole casts a $12$ -foot shadow. How tall is the flagpole, in feet?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

$\overline{AB} \parallel \overline{DE}$ , and $\overline{AE}$ and $\overline{BD}$ cross at $C$ (an hourglass, as in the lesson). If $AC = 5$ , $CE = 15$ and $AB = 4$ , find $DE$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

In $\triangle ABC$ , point $D$ is on $\overline{AC}$ so that $\angle ABD \cong \angle ACB$ . If $AB = 12$ and $AD = 8$ , find $DC$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.