SAS Similarity Rule

The SAS Similarity Rule states that if two sides of one triangle are in the same ratio as the corresponding two sides of another triangle, and the included angle between those sides is equal, then the triangles are similar.

 Side ratio + Equal included angle → Similarity

SAS (Side-Angle-Side) Similarity Rule:

If in triangles \(\triangle ABC\) and \(\triangle DEF\):

• \( \dfrac{AB}{DE} = \dfrac{AC}{DF} \)
• \( \angle A = \angle D \)

where the angle is the included angle between the two sides, then:

\( \triangle ABC \sim \triangle DEF \)

When two pairs of corresponding sides are in the same ratio and the angle between them is equal:

• The two triangles must open in the same way.
• Their overall shapes become proportional.
• The remaining angle and the third side automatically become proportional as well.

This ensures the triangles share the same shape.

The angle must lie between the two sides being compared.

In \(\triangle ABC\):

• Between AB and AC → \(\angle A\)
• Between BC and BA → \(\angle B\)
• Between AC and BC → \(\angle C\)

       A                        D
      / \                      / \
     /   \                    /   \
    B-----C                  E-----F
AB/DE = AC/DF and ∠A = ∠D → Similar

SAS similarity is used when lengths and an angle can be measured more easily than full triangle properties. Examples include:

• Surveying land using measured distances and fixed angles,
• Designing scaled models where proportions must match,
• Shadow-based height measurements,
• Architectural planning using proportional triangles.