Angle-side-angle theorem

Angle-side-angle theorem (ASA Theorem) is rule in geometry that tells when two triangles are exactly the same.[1] A triangle is a shape with three strai

Angle-side-angle theorem (ASA Theorem) is rule in geometry that tells when two triangles are exactly the same.[1] A triangle is a shape with three straight sides and three corners called angles.[1] The ASA Theorem says this: if two angles in one triangle are the same as two angles in another triangle, and the side between those two angles is also the same in both triangles, then the two triangles are congruent.[1] Congruent means the triangles are exactly the same shape and size, even if one is turned, flipped, or moved somewhere else. The side between the two angles is called the included side,[1] and it is the side that sits right between the two angles. This rule works because when two angles and the side between them are fixed, there is only one possible triangle that can be made, so both triangles must match exactly.[1] ASA is one of several triangle rules in geometry, along with SSS (three sides), SAS (two sides and the angle between them), AAS (two angles and a side that is not between them), and HL (for right triangles).[2] These rules are used to show triangles are congruent without measuring every part.[2] ASA is often used in geometry problems where shapes are drawn or split into triangles.[2] It also connects to rigid motions, which are movements like sliding, turning, or flipping a shape. Rigid motions do not change the size or shape of a figure,[3] so the triangle stays the same after moving. After two triangles are proven congruent using ASA, we can use CPCTC, which means Corresponding Parts of Congruent Triangles are Congruent. This means that after we know the triangles match, all their matching sides and angles also match.[3]

References

  1. ↑ 1.0 1.1 1.2 1.3 1.4 "2.3: The ASA and AAS Theorems". Mathematics LibreTexts. 2020-01-22. Retrieved 2026-06-21.
  2. ↑ 2.0 2.1 2.2 "Euclid's Elements, Book I, Proposition 26". mathcs.clarku.edu. Retrieved 2026-06-21.
  3. ↑ 3.0 3.1 "Angle Side Angle - Theorem, Proof, Formula & Examples". Cuemath. Retrieved 2026-06-21.

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