Finding Similar Triangles — Definition, Formula & Examples

Finding similar triangles means identifying two triangles that have the same shape but not necessarily the same size. You do this by showing that their corresponding angles are equal or that their corresponding sides are in proportion.

Two triangles are similar (denoted $\triangle ABC \sim \triangle DEF$ ) if and only if their corresponding angles are congruent and their corresponding sides are proportional. To establish similarity, it suffices to verify one of three criteria: AA (two pairs of equal angles), SAS Similarity (two pairs of proportional sides with the included angle equal), or SSS Similarity (all three pairs of sides in the same ratio).

How It Works

Start by looking for angle relationships: vertical angles, parallel lines creating alternate interior angles, or shared angles between overlapping triangles. If you can find two pairs of equal angles, the triangles are similar by AA — you don't even need side lengths. When angle information is limited, compare side lengths instead. If all three pairs of corresponding sides share the same ratio, use SSS Similarity. If two pairs of sides are proportional and the angle between them is equal, use SAS Similarity. Always write the similarity statement with vertices in corresponding order, since this determines which sides and angles match.

Worked Example

Problem: In triangle ABC, a line DE is drawn parallel to BC, where D is on AB and E is on AC. Given that AD = 4, DB = 6, and AE = 3. Find EC and show that triangle ADE is similar to triangle ABC.

Identify equal angles: Since DE is parallel to BC, angle ADE equals angle ABC (corresponding angles), and angle AED equals angle ACB. Angle A is shared by both triangles.

\angle ADE = \angle ABC, \quad \angle AED = \angle ACB

Apply AA Similarity: Two pairs of corresponding angles are equal, so the triangles are similar by AA.

\triangle ADE \sim \triangle ABC

Use the proportion to find EC: The ratio of corresponding sides must be equal. AD/AB = AE/AC, where AB = AD + DB = 10.

\frac{4}{10} = \frac{3}{AC} \implies AC = 7.5 \implies EC = AC - AE = 4.5

Answer: Triangle ADE is similar to triangle ABC by AA Similarity, and EC = 4.5.

Why It Matters

Finding similar triangles is essential for solving indirect measurement problems, such as calculating the height of a building from its shadow. It also forms the foundation for trigonometric ratios and coordinate geometry proofs, topics you will encounter repeatedly in precalculus and standardized tests like the SAT.

Common Mistakes

Mistake: Writing the similarity statement with vertices in the wrong order, such as writing △ADE ~ △CBА instead of △ADE ~ △ABC.

Correction: The order of vertices must reflect corresponding parts. If angle A corresponds to angle A, angle D corresponds to angle B, and angle E corresponds to angle C, write △ADE ~ △ABC. Incorrect order leads to wrong side pairings and incorrect proportions.