SAS Theorem — Definition, Formula & Examples

The SAS Theorem states that two triangles are congruent when two sides and the angle between them (the included angle) in one triangle are equal to two sides and the included angle in the other triangle.

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. Symbolically, if $AB \cong DE$ , $\angle B \cong \angle E$ , and $BC \cong EF$ , then $\triangle ABC \cong \triangle DEF$ .

How It Works

To apply SAS, identify two sides and the angle that lies between them in each triangle. The angle must be formed by the two sides you are comparing — this is what "included" means. If all three measurements match between the two triangles, you can conclude the triangles are congruent. Once congruence is established, every remaining side and angle in one triangle equals its corresponding part in the other (by CPCTC).

Worked Example

Problem: In triangle ABC, AB = 7 cm, BC = 10 cm, and the included angle B = 50°. In triangle DEF, DE = 7 cm, EF = 10 cm, and the included angle E = 50°. Are the triangles congruent?

Step 1: Compare the first pair of sides.

AB = DE = 7 \text{ cm}

Step 2: Check that the included angles (the angles between those sides) are equal.

\angle B = \angle E = 50°

Step 3: Compare the second pair of sides.

BC = EF = 10 \text{ cm}

Answer: Yes. By the SAS Theorem, triangle ABC is congruent to triangle DEF.

Why It Matters

SAS is one of the most frequently used congruence theorems in high school geometry proofs. Engineers and surveyors rely on it to confirm that structural components or land parcels have identical triangular shapes when direct measurement of all six parts is impractical.

Common Mistakes

Mistake: Using two sides and a non-included angle and calling it SAS.

Correction: The angle must be between the two given sides. Two sides and a non-included angle is the ambiguous SSA case, which does not prove congruence.