SSS Congruence

SSS Congruence

Side-side-side congruence. When two triangles have corresponding sides that are congruent as shown below, the triangles are congruent.

Two congruent triangles ABC and DEF with tick marks showing three pairs of equal corresponding sides (SSS).

See also

Congruence tests for triangles

Key Formula

\text{If } AB = DE,\; BC = EF,\; \text{and } AC = DF,\; \text{then } \triangle ABC \cong \triangle DEF

Where:

$AB, BC, AC$ = The three side lengths of the first triangle
$DE, EF, DF$ = The corresponding three side lengths of the second triangle
$\cong$ = The congruence symbol, meaning the triangles have equal corresponding sides and equal corresponding angles

Worked Example

Problem: Triangle PQR has sides PQ = 5 cm, QR = 7 cm, and PR = 9 cm. Triangle XYZ has sides XY = 5 cm, YZ = 7 cm, and XZ = 9 cm. Are the two triangles congruent?

Step 1: List the sides of both triangles and identify the corresponding pairs.

PQ = 5,\; QR = 7,\; PR = 9 \quad \text{and} \quad XY = 5,\; YZ = 7,\; XZ = 9

Step 2: Compare the first pair of corresponding sides.

PQ = XY = 5 \; \checkmark

Step 3: Compare the second pair of corresponding sides.

QR = YZ = 7 \; \checkmark

Step 4: Compare the third pair of corresponding sides.

PR = XZ = 9 \; \checkmark

Step 5: Since all three pairs of corresponding sides are equal, apply the SSS Congruence rule.

\triangle PQR \cong \triangle XYZ \quad \text{by SSS}

Answer: Yes, triangle PQR is congruent to triangle XYZ by SSS Congruence.

Another Example

This example shows SSS applied within a larger figure where two triangles share a common side (the diagonal). Recognizing shared sides is a key skill in geometry proofs.

Problem: In quadrilateral ABCD, a diagonal BD divides it into two triangles. Given AB = 6, BD = 8, AD = 10, BC = 6, and CD = 10. Prove that triangle ABD is congruent to triangle CBD.

Step 1: Identify the sides of triangle ABD.

AB = 6,\; BD = 8,\; AD = 10

Step 2: Identify the sides of triangle CBD. Note that BD is a shared side between the two triangles.

CB = 6,\; BD = 8,\; CD = 10

Step 3: Match corresponding sides: AB corresponds to CB, BD is common to both, and AD corresponds to CD.

AB = CB = 6,\; BD = BD = 8,\; AD = CD = 10

Step 4: All three pairs of corresponding sides are equal, so apply SSS.

\triangle ABD \cong \triangle CBD \quad \text{by SSS}

Answer: Triangle ABD is congruent to triangle CBD by SSS Congruence.

Frequently Asked Questions

What is the difference between SSS and SAS congruence?

SSS requires all three pairs of corresponding sides to be equal, with no angle information needed. SAS (Side-Angle-Side) requires two pairs of corresponding sides to be equal and the angle between those two sides (the included angle) to also be equal. Both prove congruence, but they use different sets of information.

Does SSS work for proving similarity as well as congruence?

Yes, there is a related rule called SSS Similarity. Instead of requiring the sides to be equal, it requires all three pairs of corresponding sides to be in the same ratio (proportional). If the sides are proportional but not equal, the triangles are similar but not congruent.

Why does SSS guarantee the angles are equal too?

Once you fix the three side lengths of a triangle, there is only one possible triangle shape you can build (up to reflection and rotation). This is because the side lengths completely determine the angles through the law of cosines. So if two triangles share all three side lengths, their angles must match as well.

SSS Congruence vs. SAS Congruence

	SSS Congruence	SAS Congruence
What you need to know	All three pairs of corresponding sides	Two pairs of corresponding sides and the included angle
Angle information required	None	One angle (must be between the two known sides)
When to use	When you know all side lengths but no angles	When you know two sides and the angle between them
Common scenario	Coordinate geometry where distances are computed	Problems where an angle measurement is given or proven equal

Why It Matters

SSS Congruence is one of the first proof techniques you learn in geometry and appears frequently on standardized tests and in homework problems involving triangle proofs. It is especially useful in coordinate geometry, where you can calculate all side lengths using the distance formula and then conclude congruence without ever measuring angles. Engineers and architects also rely on this principle — once three side lengths of a triangular structure are fixed, its shape is rigid and fully determined.

Common Mistakes

Mistake: Matching sides incorrectly by comparing the longest side of one triangle to the shortest side of another.

Correction: Always match corresponding sides carefully. Corresponding sides are opposite corresponding angles. Write out the correspondence clearly (e.g., triangle ABC ≅ triangle DEF means AB↔DE, BC↔EF, AC↔DF) before comparing lengths.

Mistake: Trying to use SSS with non-triangular polygons, such as quadrilaterals.

Correction: SSS only works for triangles. Knowing all four side lengths of a quadrilateral does not determine its shape — a quadrilateral can flex into different shapes with the same side lengths. For other polygons, you need additional information like angles or diagonals.

Related Terms

Congruence Tests for Triangles — Overview of all triangle congruence rules
Congruent — General definition of congruence in geometry
Triangle — The shape SSS Congruence applies to
Side of a Polygon — The line segments compared in SSS
Corresponding — How to identify matching sides and angles
SAS Congruence — Congruence using two sides and included angle
ASA Congruence — Congruence using two angles and included side
Distance Formula — Used to compute side lengths for SSS proofs