Side Angle Side (SAS) — Congruence and Similarity Rules

TL;DR

Side Angle Side (SAS) is one criterion that does two jobs: two sides and the included angle prove triangles congruent when the sides are equal, and similar when the sides are proportional. This article keeps the two apart — same angle condition, different side condition — with a labelled diagram and proof for each, plus worked examples and the mistakes that blur them.

What Does Side Angle Side Mean?

Side Angle Side names a specific arrangement of three parts: two sides and the angle that sits between them. That middle angle is called the included angle — it is formed by the two named sides meeting at a vertex. The phrase "side, angle, side" is a memory aid for the order: a side, then the angle in the corner, then the second side.

This arrangement is powerful because it locks a triangle. Fix two side lengths and the angle between them, and there is exactly one way to close the triangle — the third side and the other two angles are forced. That single fact powers both versions of the rule.

Note the symbols up front: congruent uses ≅ (identical), similar uses ∼ (same shape, scalable). Keeping these straight is half the battle.

The SAS Congruence Rule

The rule: If two sides and the included angle of one triangle are equal to two sides and the included angle of another, the triangles are congruent.

In symbols, for △ABC and △DEF:

AB=DE,∠A=∠D,AC=DF;⇒;△ABC≅△DEF

The variable glossary: AB and AC are the two sides meeting at vertex A; ∠A is the included angle between them.

Why it holds. Place △DEF on top of △ABC so vertex D lands on A. Because ∠D=∠A, side DE falls along AB; because DE=AB, point E lands exactly on B. The same logic puts F on C, so every part coincides and the triangles are congruent.

The SAS Similarity Rule

The rule: If two sides of one triangle are proportional to two sides of another, and the included angles are equal, the triangles are similar.

In symbols:

AB/DE = AC/DF,∠A=∠D;⇒;△ABC∼△DEF

The included angle still has to match.

Why it holds. Equal angles set the shape at the shared vertex; the equal ratio of the two adjacent sides means the second triangle is the first one enlarged (or shrunk) by that ratio — a scale factor.

How Do You Tell SAS Congruence From SAS Similarity?

A reader question that comes up constantly: if both need the included angle, what actually separates them? The side condition, and nothing else.

Sides equal (AB=DE): the triangles are the same size → congruence, ≅.
Sides in a common ratio (AB/DE, AC/DF, ratio not 1): the triangles are scaled copies → similarity, ∼.

Examples of Side Angle Side Congruence and Similarity

Example 1

In △PQR and △XYZ: PQ=XY=6 cm, ∠Q=∠Y=48°, QR=YZ=9 cm. Congruent or similar — and by what rule?

The sides are equal, and the angle ∠Q sits between PQ and QR (included).

Final answer: congruent by SAS.

Example 2

In △ABC and △DEF: AB=4, DE=8, AC=5, DF=10, and ∠A=∠D=60°. Are they congruent?

Final answer: similar, not congruent.

Example 3

Prove that in isosceles △ABC with AB=AC, the angle bisector AD splits it into two congruent triangles.

Final answer: congruent by SAS.

Example 4

△ABC has AB=3 cm, AC=4 cm, ∠A=40°. △DEF has ∠D=40°, DE=9 cm, DF=12 cm. Find EF given BC=5 cm.

Final answer: EF=15 cm.

Example 5

Two map plots share a corner. From that corner, plot 1 runs 30 m and 40 m with a 70° angle between; plot 2 runs 60 m and 80 m with the same 70° angle. The diagonal of plot 1 is 47 m. What is the diagonal of plot 2?

Final answer: 94 m.

Example 6

In △ABC, point D lies on AB and E on AC so that AD/AB = AE/AC = 1/3. Prove △ADE∼△ABC.

Final answer: similar by SAS.

Why One Rule Carries Both Jobs

The reason SAS does double duty is structural.

The included angle fixes the shape at the corner.
Engineering and design run on the similarity half.
Congruence is the trustworthy floor of proof.

Where SAS goes sideways

Mistake 1: Using a non-included angle

Where it slips in: When the angle in the data is not the one between the two named sides.

The correct way: Confirm the angle is wedged between the two sides before invoking SAS.

Mistake 2: Mixing up equal and proportional

Where it slips in: Reading a similarity problem but reaching for the congruence verdict, or vice versa.

The correct way: Test the sides numerically first.

Mistake 3: Forgetting the common side or angle

Where it slips in: Proofs where two triangles share a side or a vertex angle.

The correct way: Write the shared part explicitly — AD=AD or ∠A=∠A.

Conclusion

Side Angle Side uses two sides and the included angle for multiple purposes.
SAS congruence needs the sides equal and proves triangles identical (≅).
SAS similarity needs the sides proportional and proves triangles are scaled copies (∼).
The included angle must match in both; only the side condition switches.