--- title: "Triangle congruence criteria" description: "Know and use the criteria for triangle congruence (SSS, SAS, ASA, RHS), use standard labelling conventions for sides and angles of triangle ABC, and determine whether two triangles are congruent" canonical: https://lightmysky.com/learn/mathematics/triangle-congruence-criteria-mt_167X6Ax8P7 source: https://lightmysky.com/learn/mathematics/triangle-congruence-criteria-mt_167X6Ax8P7.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Triangle congruence criteria Know and use the criteria for triangle congruence (SSS, SAS, ASA, RHS), use standard labelling conventions for sides and angles of triangle ABC, and determine whether two triangles are congruent Subject: Mathematics · Area: Geometry · Ages 12 to 14 Page: https://lightmysky.com/learn/mathematics/triangle-congruence-criteria-mt_167X6Ax8P7 ## Ready when they can - State which congruence criterion (SSS, SAS, ASA, or RHS) applies to a given pair of triangles - Use the notation △ABC ≅ △DEF and match corresponding sides and angles - Determine whether given measurements produce a unique triangle, more than one, or none ## Lesson: Proving triangles congruent Every triangle has three vertices, named with capital letters like A, B, and C. Each side gets the matching lowercase letter of the vertex across from it. Side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. Two triangles are congruent when they are exactly the same size and shape, even if one is turned or flipped. In symbols that is △ABC ≅ △DEF. The letter order matters: A matches D, B matches E, C matches F, so side AB equals side DE. *(drawing: Each capital vertex letter pairs with the lowercase side letter across from it.)* **Example.** Triangle ABC has sides 5 cm, 6 cm, and 7 cm. Triangle DEF has sides 5 cm, 6 cm, and 7 cm too, matched in the same order. All three pairs of sides match, so the triangles are congruent by SSS (side-side-side). You do not even need to check the angles. Four checks prove congruence. SSS: all three sides match. SAS: two sides and the angle between them match. ASA: two angles and the side between them match. RHS: in a right triangle, the right angle, the hypotenuse, and one other side match. For SAS and ASA, the letter in the middle of the name must sit between the other two. RHS is different: its letters do not follow the between rule. *(drawing: Given two sides and a non-included angle, the side opposite the angle swings and can land on the ray in 0, 1, or 2 places.)* Three side lengths that close up give exactly one triangle, which is why SSS works. The triangle inequality must hold: each side must be shorter than the other two added together. If one length is longer than or equal to the other two added together, no triangle can be built at all. Equality gives a degenerate flat triangle. Two sides with an angle that is not between them do not always fix a single triangle: sometimes two different triangles fit, sometimes one, sometimes none. That is why SSA is not a general criterion, though RHS is a special case that does give uniqueness. Three matching angles fix the shape but not the size, so the triangles can be different sizes. **Recap.** Match corresponding vertices in order, then check SSS, SAS, ASA, or RHS to prove two triangles are congruent. ## Practice 17 questions on this page, each with its working shown. ## Needs first - [Accurate drawing and scale diagrams](https://lightmysky.com/learn/mathematics/accurate-drawing-and-scale-diagrams-mt_Jf8xcX4UTq) - [Geometric terms and notation](https://lightmysky.com/learn/mathematics/geometric-terms-and-notation-mt_WtIFJSCQIT) ## Opens up - [Congruent Triangles and the Congruence Conditions](https://lightmysky.com/learn/mathematics/congruent-triangles-and-the-congruence-conditions-mt_it8t7mY71J)