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SAS congruence without mixing the angle

2026-09-26Plane GeometryMs Ho

Two sides and the included angle — the angle must sit between those sides.

Key formulas

SAS, ASA, SSS, RHS

Included means between

If you know AB, AC and ∠B, that is not SAS. You need the angle at A between AB and AC.

RHS for right triangles

Right angle, hypotenuse and one other side give congruence — a special case worth memorising.

After congruence

Corresponding parts are equal. State which vertices match before transferring lengths.

Worked example

Worked example

Problem. In △ABC and △DEF, AB = DE = 6, AC = DF = 7, and ∠A = ∠D = 50°. Are the triangles congruent?

Solution. The equal angle is included between the two equal sides (SAS). Yes — △ABC ≅ △DEF by SAS.

SSA warning

Two sides and a non-included angle is not a congruence test in general.

Review quiz

Knowing AB=DE, AC=DF and ∠B=∠E (not at A) is generally:

Practice questions

  1. Explain why SSA fails with a counter-sketch.
  2. List the four standard congruence tests used in secondary geometry.

Study notes

Can SAS prove similarity only?

SAS with proportional sides is a similarity test; equal sides give congruence.

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