SAS Congruence Rule

The SAS Congruence Rule states that if two sides and the included angle of one triangle are equal to the two corresponding sides and the included angle of another triangle, then the triangles are congruent.

The word “included angle” is important — it is the angle between the two sides.

 Two sides + the angle between them → Congruence

SAS (Side-Angle-Side) Rule:

If in two triangles, two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of the other, then the triangles are congruent.

Symbolically, if in \(\triangle ABC\) and \(\triangle DEF\):

• \(AB = DE\)
• \(BC = EF\)
• \(\angle B = \angle E\)

then:

\(\triangle ABC \cong \triangle DEF\)

When two sides and the angle between them are fixed:

• The triangle’s exact shape is determined.
• No other triangle can be formed with different angles or side lengths.
• The included angle ensures the sides cannot swing or change shape.

This removes any ambiguity and guarantees congruence.

The included angle is the angle between the two sides being compared. It must lie between the given sides.

For example, in \(\triangle ABC\):

• Between sides AB and AC → included angle is \(\angle A\)
• Between sides AB and BC → included angle is \(\angle B\)
• Between sides BC and CA → included angle is \(\angle C\)

       A                   D
      /|                  /|
     / |                 / |
    /  |                /  |
   B---C               E---F
AB = DE, BC = EF, ∠B = ∠E → Congruent

The SAS rule appears in situations where two sides are fixed and connected by a fixed angle:

• Mechanical linkages with rods connected at fixed angles,
• Roof frames where two beams meet at a fixed angle,
• Geometric patterns made with fixed corner joints,
• Wooden or metal triangular components cut using angle templates.