Side-angle-side theorem

Side-angle-side theorem (SAS Theorem) is rule in geometry used to show when two triangles are congruent.[1] Congruent triangles are triangles that have

Side-angle-side theorem

Side-angle-side theorem (SAS Theorem) is rule in geometry used to show when two triangles are congruent.[1] Congruent triangles are triangles that have exactly the same size and shape, even if one triangle is turned, flipped, or moved somewhere else.[1] The SAS Theorem says that 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.[1] The included angle is the angle that is formed between the two sides being compared. Congruent sides are sides that have the same length, and congruent angles are angles that have the same measure, meaning they open the same amount.[1] This theorem is important because it gives a way to prove that two triangles are exactly the same without checking every side and every angle. It is one of several triangle congruence rules, along with SSS (Side-Side-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), and HL (Hypotenuse-Leg).[2] All of these rules help show when triangles match perfectly in size and shape.[2] The SAS Theorem works because if two sides and the angle between them are fixed, then the triangle cannot change shape without breaking those conditions, so only one exact triangle can be made. This makes SAS a strong and reliable way to prove triangle congruence. It is often used in geometry problems and proofs, especially when working with shapes made from line segments, transformations, or rigid motions.[3] Rigid motions are movements like rotations, reflections, and translations that preserve distance and angle measure, meaning they keep figures congruent.[3] SAS also connects to CPCTC, which means that once two triangles are proven congruent, their corresponding parts are also congruent.

References

  1. ↑ 1.0 1.1 1.2 1.3 "Side-angle-side theorem | Definition, Congruence, & Facts | Britannica". Encyclopedia Britannica. Retrieved 2026-06-21.
  2. ↑ 2.0 2.1 "Euclid's Elements, Book I, Proposition 26". mathcs.clarku.edu. Retrieved 2026-06-21.
  3. ↑ 3.0 3.1 "6.8: The SAS Theorem". Mathematics LibreTexts. 2025-11-04. Retrieved 2026-06-21.

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