Triangles [Congruency in Triangles]

211. SAS: The Rigid Triangle · Two sides and the included angle lock the shape

The two sides and the included angle always match, so triangle DEF reshapes right along with ABC and stays perfectly congruent.

By the SAS rule (Side–Angle–Side), if two sides and the included angle — the angle that sits between those two sides — of one triangle equal the matching parts of another, the triangles are congruent. Once two sides and the angle between them are fixed, the third side and the other two angles are forced: the triangle is rigid, so its shape and size cannot change.

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Selina ICSE: Triangles [Congruency in Triangles]

What this lesson covers

Try to break it

Drag A around B on its circle (radius = DE) and C around B on its circle (radius = EF). AB and BC stay locked equal to DE and EF. There's only one configuration where ∠ABC equals ∠DEF — and at that configuration, △ABC is congruent to △DEF. SAS: two sides and the included angle pin the whole triangle.

How you build it

Build a triangle from two sides and the included angle (SAS).

Point tool: mark point E — the corner that will hold the included angle.
Point tool: mark point F to the right of E.
Segment tool: join E to F. This is the first given side.
Angle tool: set the dial to the included angle (say 60°), click E, then click toward F. The arm is the direction of side ED.
Point tool: mark point D out along the arm — ED is the second side, and ∠E sits between EF and ED.
Segment tool: join D to F. Triangle DEF — your given triangle — is complete.
Now copy it by SAS. Point tool: mark point B to the right — the corner of triangle ABC.
Ray tool: click B, then click further right — side BC lies on this ray.
Arc tool: click centre E, drag until the arc snaps to F, locking the radius to side EF.
Arc tool: click centre B — the arc crosses the base ray at C, so BC = EF.
Point tool: mark C where the arc meets the base ray.
Angle tool: set the dial to the SAME value you used at E, click B, then click toward C. The arm is the direction of side BA.
Arc tool: click centre E, drag until the arc snaps to D, locking the radius to side ED.
Arc tool: click centre B — the arc crosses the angle arm at A, so BA = ED.
Point tool: mark A where the arc meets the arm.
Segment tool: join A to C. Two sides (BC = EF, BA = ED) and the included angle (∠B = ∠E) match — so triangle ABC ≅ triangle DEF by SAS.

The proof, step by step

Prove that two sides and the included angle make the triangles congruent (SAS).

In Δ ABC and Δ DEF, AB = DE (Given)
∠ABC = ∠DEF (Given)
BC = EF (Given)
∴ Δ ABC ≅ Δ DEF (By SAS Congruence Rule)

Worked example

In Δ PQR and Δ XYZ, PQ = XY, QR = YZ, and ∠Q = ∠Y. Which congruence criterion proves Δ PQR ≅ Δ XYZ?

Since two sides and the included angle of Δ PQR are equal to the corresponding two sides and included angle of Δ XYZ, the triangles are congruent by the SAS (Side-Angle-Side) criterion.

SSS
SAS — correct
ASA
RHS