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Similar Triangle Theorems — Definition, Formula & Examples

Similar Triangle Theorems are the rules that determine when two triangles are similar — meaning they have the same shape but not necessarily the same size. The three main theorems are AA (Angle-Angle), SAS (Side-Angle-Side) Similarity, and SSS (Side-Side-Side) Similarity.

Two triangles are similar if and only if their corresponding angles are congruent and their corresponding sides are proportional. The Similar Triangle Theorems provide sufficient conditions to establish this relationship: (1) AA Similarity — if two angles of one triangle are congruent to two angles of another, the triangles are similar; (2) SAS Similarity — if two sides of one triangle are proportional to two sides of another and the included angles are congruent, the triangles are similar; (3) SSS Similarity — if all three pairs of corresponding sides are proportional, the triangles are similar.

Key Formula

aa′=bb′=cc′=k\frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'} = k
Where:
  • a,b,ca, b, c = Side lengths of the first triangle
  • a′,b′,c′a', b', c' = Corresponding side lengths of the second triangle
  • kk = Scale factor (ratio of similarity)

How It Works

To use a similarity theorem, you compare parts of two triangles and check whether one of the three conditions is met. For AA, find two pairs of equal angles — the third pair must also be equal by the angle sum property. For SAS, verify that two pairs of sides share the same ratio and the angle between those sides is congruent. For SSS, compute the ratios of all three pairs of corresponding sides and confirm they are equal. Once similarity is established, every pair of corresponding sides shares a common scale factor kk, so you can set up proportions to find unknown side lengths.

Worked Example

Problem: Triangle ABC has sides AB = 6, BC = 9, and AC = 12. Triangle DEF has sides DE = 4, EF = 6, and DF = 8. Determine whether the triangles are similar, and if so, find the scale factor.
Step 1: Pair corresponding sides from smallest to smallest. The shortest sides are AB = 6 and DE = 4. The medium sides are BC = 9 and EF = 6. The longest sides are AC = 12 and DF = 8.
Step 2: Compute the ratio of each pair of corresponding sides.
ABDE=64=32,BCEF=96=32,ACDF=128=32\frac{AB}{DE} = \frac{6}{4} = \frac{3}{2}, \quad \frac{BC}{EF} = \frac{9}{6} = \frac{3}{2}, \quad \frac{AC}{DF} = \frac{12}{8} = \frac{3}{2}
Step 3: All three ratios are equal, so by the SSS Similarity Theorem the triangles are similar.
Answer: Triangle ABC is similar to triangle DEF by SSS Similarity, with a scale factor of k = 3/2.

Another Example

Problem: In the figure, angle P = 50° and angle Q = 70° in triangle PQR. In triangle XYZ, angle X = 50° and angle Z = 60°. Are the triangles similar?
Step 1: Find the missing angle in each triangle using the angle sum property (angles add to 180°).
∠R=180°−50°−70°=60°,∠Y=180°−50°−60°=70°\angle R = 180° - 50° - 70° = 60°, \quad \angle Y = 180° - 50° - 60° = 70°
Step 2: List the angles of each triangle in order: Triangle PQR has 50°, 70°, 60°. Triangle XYZ has 50°, 70°, 60°.
Step 3: Two pairs of angles are equal (in fact all three are), so by the AA Similarity Theorem, triangle PQR ~ triangle XYZ.
Answer: Yes. Triangle PQR ~ Triangle XYZ by AA Similarity, with corresponding angles P↔X, Q↔Y, R↔Z.

Why It Matters

Similar triangle theorems appear throughout high school geometry and are essential on standardized tests like the SAT. In trigonometry, the very definition of sine, cosine, and tangent relies on the fact that similar right triangles produce consistent ratios. Professionals in surveying, architecture, and engineering routinely use triangle similarity to calculate distances and heights that cannot be measured directly.

Common Mistakes

Mistake: Confusing SAS Similarity with SAS Congruence. Students check that two sides are equal (not proportional) and an included angle is congruent, then claim similarity.
Correction: For SAS Similarity, the two pairs of sides must be proportional — sharing the same ratio — not necessarily equal. Equal sides would prove congruence, not just similarity.
Mistake: Matching corresponding sides incorrectly. Students pair the first-listed side of one triangle with the first-listed side of the other without checking which sides actually correspond.
Correction: Always match sides by their position relative to equal angles. The side opposite the largest angle in one triangle corresponds to the side opposite the largest angle in the other.

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