# Side Angle Side (SAS) — Congruence and Similarity Rules

[#Geometry](/content/tag/geometry/index.html)

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](/content/math/geometry/scale-factor/index.html).

## 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.