SAS criterion

You do not need to check all three sides. Two sides and the included angle (the angle between them) are enough.

Idea

SAS (Side-Angle-Side) criterion. If two sides of one triangle and the included angle (the angle formed at the vertex where the two sides meet) are equal to the corresponding parts of another triangle, then the two triangles are congruent.

In symbols: if $AB = DE$, $\angle A = \angle D$, $AC = DF$, then $\triangle ABC \cong \triangle DEF$.

The key word is included. The angle must be between the two named sides. If the angle is not included, the criterion fails.

Why does it work? Two sides and the angle between them fully determine the triangle: from the angle's vertex, both arms have fixed lengths, so the third side (the one connecting the far ends of the arms) has a single possible length. The whole shape is locked in.

Common pitfall: SSA. Two sides and an angle that is not included is called SSA (or ASS). It does not guarantee congruence , two different triangles can share these three pieces.

Worked examples

Example 1. In $\triangle ABC$, $AB = 6$, $\angle A = 50^\circ$, $AC = 8$. In $\triangle PQR$, $PQ = 6$, $\angle P = 50^\circ$, $PR = 8$. Congruent?

By SAS (with the angle included between the named sides), yes.

Example 2. Two sides $5, 7$ and the angle between them $60^\circ$ , uniquely determine a triangle?

Yes. The third side is fixed (you can compute it using the cosine rule in higher classes; here we just accept that it is fixed). SAS guarantees uniqueness.

Example 3. Why does SSA fail?

Imagine fixing one side and an angle at one end, and a fixed second-side length. The second side, swung from its endpoint, may hit the base in two different points , giving two different triangles. So SSA is ambiguous.

Example 4. State SAS in your own words.

If you know two sides of a triangle and the angle exactly between them, the triangle is determined.

Try it yourself

1. State SAS in symbols.
2. $\triangle ABC$: $AB=4, BC=5, \angle B=70^\circ$. $\triangle XYZ$: $XY=4, YZ=5, \angle Y=70^\circ$. Congruent?
3. Two triangles: sides $5, 6$ included angle $90^\circ$ each. Congruent?
4. SSA , give an example showing it can fail.
5. SAS or SSS , which uses only sides?
6. Two triangles share two sides and an angle that is not between them. Definitely congruent?
7. Sketch a triangle given $AB=5$, $\angle A=45^\circ$, $AC=6$. How many such triangles?
8. Why is SAS strong while AAA (just angles) is not?

Activity

Take two straws of $5$ cm and $7$ cm. Fix them at one end with tape, making any angle. Then connect their far ends with a third straw. The shape is locked once the angle is set. Now repeat for a friend with the same angle and the same straws , both triangles will be congruent.