ASA, SAS, and SSS Criteria
Aligned to G-CO.B.8 — Common Core State Standards for Mathematics.
What this lesson teaches
The shortcut criteria follow from the rigid-motion definition of congruence. SAS: two sides and the included angle fix a triangle's shape, so a rigid motion maps one onto another with matching parts. ASA and SSS work the same way. These let you prove congruence without checking all six parts.
Worked example
Prove two triangles are congruent by SAS. Triangle ABC has AB = 6, angle A = 50°, AC = 8. Triangle DEF has DE = 6, angle D = 50°, DF = 8. The angle sits between the two given sides in each triangle (included angle). Place A on D and align side AB with DE; the 50° angle forces AC to align with DF, and equal lengths make B land on E and C on F. So ABC = DEF by SAS.
Practice questions
- Which criterion (ASA, SAS, or SSS) proves congruence when you know all three side lengths match?
- Two triangles share: angle = 40°, side = 5, angle = 70°, with the side between the angles. Name the criterion.
- Explain why SSA (two sides and a non-included angle) is not a valid congruence criterion.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Criterios ALA, LAL y LLL
Los criterios abreviados se derivan de la definición de congruencia por movimientos rígidos. LAL: dos lados y el ángulo incluido fijan la forma de un triángulo, así que un movimiento rígido mapea uno sobre otro con partes coincidentes. ALA y LLL funcionan igual. Permiten demostrar congruencia sin revisar las seis partes.
Ejemplo: Demuestra que dos triángulos son congruentes por LAL. El triángulo ABC tiene AB = 6, ángulo A = 50°, AC = 8. El triángulo DEF tiene DE = 6, ángulo D = 50°, DF = 8. El ángulo está entre los dos lados dados en cada triángulo (ángulo incluido). Coloca A sobre D y alinea AB con DE; el ángulo de 50° obliga a AC a alinearse con DF, y las longitudes iguales hacen que B caiga en E y C en F. Así ABC = DEF por LAL.
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