SAS Congruence

High School

Definition

Stands for side-angle-side congruence. If two of a pair of triangles' corresponding sides and one of their included angles are the same, you can use SAS to prove that the two are congruent.

Worked examples

\(\triangle ABC \cong \triangle DEF\) given \(AB = DE\), \(\angle A = \angle D\), \(AC = DF\)

Two sides and the included angle between them match, so SAS proves the triangles congruent.

\(\triangle PQR\) has \(PQ = 5\), \(\angle P = 40^\circ\), \(PR = 7\); \(\triangle XYZ\) has \(XY = 5\), \(\angle X = 40^\circ\), \(XZ = 7\)

The angle must be between the two given sides for SAS to apply — here it is, so they are congruent.

Common mistakes

\(AB = DE\), \(BC = EF\), \(\angle C = \angle F\) → SAS → The angle must be included between the two sides If the angle is not between the matched sides, SAS does not apply; you need SSS or AAS instead.
\(AB = DE\), \(\angle B = \angle E\), \(AC = DF\) → SAS → \(AB = DE\), \(\angle A = \angle D\), \(AC = DF\) → SAS The angle must be between the two sides; \(\angle B\) is not between \(AB\) and \(AC\).
Using SAS with two sides and any angle → SAS requires the angle to be the included angle between the two sides Side-Side-Angle (SSA) is not a valid congruence shortcut; only the included angle works for SAS.

Where you'll use it next

You'll use SAS to prove triangles congruent in geometry proofs, then apply congruence to solve for unknown sides and angles, prove properties of quadrilaterals, and build toward similarity and trigonometry.

Found in 1 StudyPug lesson

Triangle Congruence: Mastering SAS and HL Proofs

Geometry

Dive into the world of triangle congruence! Learn to apply SAS and HL proofs with confidence. Boost your geometry skills and excel in problem-solving with our expert-led lessons and practice exercises.

10th Grade10th