Similarity of Triangles

Two triangles are said to be similar when they have exactly the same shape but are not necessarily the same size. This means their corresponding angles are equal and their corresponding sides are proportional. Similarity is a central idea in Euclidean geometry and forms the backbone of trigonometry, coordinate geometry, and countless real-world applications such as mapmaking, architecture, and engineering scale models. Formally, if triangle ABC is similar to triangle DEF, written △ABC ∼ △DEF, then ∠A = ∠D, ∠B = ∠E, ∠C = ∠F, and the ratios AB/DE = BC/EF = CA/FD all equal the same constant k, called the ratio of similarity or scale factor. When k = 1 the triangles are congruent, making congruence a special case of similarity. Unlike congruence, which requires six pieces of matching information (three sides and three angles), similarity can be established with far fewer conditions. Mathematicians have identified three main criteria — AA, SAS~, and SSS~ — each providing a shortcut to prove that two triangles are similar without checking all six correspondences. Understanding these criteria, why they work, and how to apply them is the focus of this article.

Key takeaways

AA (Angle-Angle) Criterion
SAS~ (Side-Angle-Side Similarity) Criterion
SSS~ (Side-Side-Side Similarity) Criterion
Using the Scale Factor to Find Missing Lengths
Common Configurations Producing Similar Triangles

AA (Angle-Angle) Criterion

If two angles of one triangle are equal to two corresponding angles of another triangle, the triangles are similar. Because the angle sum in any triangle is 180°, once two pairs of angles match, the third pair must automatically match as well. This is the most frequently used and easiest criterion to apply. Example: if ∠A = ∠D = 50° and ∠B = ∠E = 70°, then ∠C = ∠F = 60°, so △ABC ∼ △DEF by AA.

SAS~ (Side-Angle-Side Similarity) Criterion

If two pairs of corresponding sides are proportional AND the included angles (the angles between those sides) are equal, the triangles are similar. The tilde (~) distinguishes this from the congruence version (SAS). Formally: if AB/DE = AC/DF and ∠A = ∠D, then △ABC ∼ △DEF. It is critical that the equal angle is included between the two proportional sides; otherwise the criterion does not hold.

SSS~ (Side-Side-Side Similarity) Criterion

If all three pairs of corresponding sides are proportional, the triangles are similar, regardless of any angle information. Formally: if AB/DE = BC/EF = CA/FD, then △ABC ∼ △DEF. This criterion guarantees the angles will automatically match. For instance, sides 3, 4, 5 and 6, 8, 10 form similar triangles because 3/6 = 4/8 = 5/10 = 1/2.

Using the Scale Factor to Find Missing Lengths

Once similarity is established, the scale factor k = (side of one triangle)/(corresponding side of other triangle) lets you find any unknown length. If △ABC ∼ △DEF with scale factor k = 3/2, then every side of △DEF is 2/3 of the corresponding side of △ABC. Areas scale by k², so if the area of △ABC is 18 cm², the area of △DEF is 18 × (2/3)² = 8 cm².

Common Configurations Producing Similar Triangles

Several geometric setups automatically produce similar triangles: (1) A line drawn parallel to one side of a triangle cuts the other two sides proportionally and creates a smaller triangle similar to the original (Basic Proportionality Theorem / Thales's theorem). (2) The altitude drawn from the right angle of a right triangle to the hypotenuse creates two smaller triangles each similar to the original and to each other. (3) Two triangles formed by intersecting chords or secants in a circle are often similar by AA.

The Key Equation

\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} = k

AB, BC, CA — Sides of triangle ABC (length (e.g., cm)): The three side lengths of the first triangle, measured in order of the stated correspondence.
DE, EF, FD — Sides of triangle DEF (length (e.g., cm)): The three side lengths of the second triangle, each corresponding to the side listed directly above it in the ratio.
k — Scale factor (dimensionless): The common ratio of corresponding sides. If k > 1 the second triangle is smaller; if k < 1 it is larger; if k = 1 the triangles are congruent.

Worked Example

Problem. In the diagram, DE is parallel to BC in triangle ABC. AD = 4 cm, DB = 6 cm, DE = 5 cm, and AE = 3 cm. (a) Prove that △ADE ∼ △ABC. (b) Find BC and EC.

BC = 12.5\text{ cm},\quad EC = 4.5\text{ cm}

Answer. BC = 12.5 cm and EC = 4.5 cm

Figure

Diagram

Origins · 1 minThales of Miletus and how this was discovered

Thales of Miletus

-624–-546 · Greek

Thales is credited with several foundational results in geometry, including the theorem bearing his name: a line drawn parallel to one side of a triangle divides the other two sides proportionally. This result is the geometric engine behind the AA similarity criterion and the concept of similar triangles. He reportedly used similar triangles to measure the height of the Egyptian pyramids by comparing the shadow of a stick to the shadow of the pyramid at the moment when a person's shadow equals their height. Ancient sources claim Thales predicted a solar eclipse in 585 BCE, making him one of the earliest known scientists to apply mathematical reasoning to natural phenomena.

Congruence of Triangles — Congruence is the special case of similarity where the scale factor k = 1; understanding both criteria side by side reinforces the distinction between same shape and same size.
Basic Proportionality Theorem (Thales's Theorem) — This theorem states that a line parallel to one side of a triangle divides the other two sides proportionally, and is the primary tool for proving AA similarity in many problems.
Trigonometric Ratios — Sine, cosine, and tangent are defined using the sides of right triangles; their consistency across all similar right triangles is exactly why they depend only on the angle, not the triangle's size.
Pythagorean Theorem — One elegant proof of the Pythagorean theorem uses the fact that the altitude to the hypotenuse of a right triangle creates two triangles each similar to the original.