SSS Congruence

Learn to prove triangles congruent with the SSS (Side-Side-Side) criterion, match corresponding sides and check SSS on a coordinate grid.

Intermediate25 min18 problems
Practice Problems

Definition

The SSS (Side-Side-Side) Congruence Criterion says: if the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent.
If AB‾≅DE‾,BC‾≅EF‾,CA‾≅FD‾\text{If } \overline{AB} \cong \overline{DE}, \quad \overline{BC} \cong \overline{EF}, \quad \overline{CA} \cong \overline{FD}
Then △ABC≅△DEF\text{Then } \triangle ABC \cong \triangle DEF
You do not need to measure a single angle. Three side lengths fix a triangle completely: this is called triangle rigidity. Once the three sides are chosen, the angles are forced, so every triangle with those sides is the same triangle, only moved, turned or flipped.
Matching the sides: the order of the letters in △ABC≅△DEF\triangle ABC \cong \triangle DEF tells you which parts correspond: A↔DA \leftrightarrow D, B↔EB \leftrightarrow E, C↔FC \leftrightarrow F. Pair the shortest side with the shortest, the middle with the middle and the longest with the longest, then name the vertices in the matching order.
On a coordinate grid you find each side with the distance formula:
d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Try it now

Which information is enough to prove two triangles congruent by SSS?

Worked Examples

In △ABC\triangle ABC: AB=5AB = 5 cm, BC=7BC = 7 cm, CA=6CA = 6 cm. In △DEF\triangle DEF: DE=5DE = 5 cm, EF=7EF = 7 cm, FD=6FD = 6 cm. Are the triangles congruent?
1

Compare the first pair of sides

AB=DE=5AB = DE = 5 cm → AB‾≅DE‾\overline{AB} \cong \overline{DE}
2

Compare the second pair

BC=EF=7BC = EF = 7 cm → BC‾≅EF‾\overline{BC} \cong \overline{EF}
3

Compare the third pair

CA=FD=6CA = FD = 6 cm → CA‾≅FD‾\overline{CA} \cong \overline{FD}
4

Apply SSS

All three pairs of sides are congruent → △ABC≅△DEF\triangle ABC \cong \triangle DEF by SSS

Common Mistakes

Writing the vertices in the wrong order, like △PQR≅△XYZ\triangle PQR \cong \triangle XYZ when PP matches YY

Why it's wrong: A congruence statement is a promise that the first letters correspond, the second letters correspond and the third letters correspond. Alphabetical order is not the same as matching order.

Correct: Pair the sides by length first, then find which vertices the matching sides meet at. Write the vertices in that order: △PQR≅△YZX\triangle PQR \cong \triangle YZX.

Thinking SSS also works for quadrilaterals

Why it's wrong: A square with sides 4 and a thin rhombus with sides 4 have four equal pairs of sides, but they are not congruent. Four sides do not fix the angles.

Correct: Side lengths alone only fix the shape of a triangle. For quadrilaterals you also need angles or a diagonal.

Forgetting the shared side in a proof

Why it's wrong: When two triangles share a side, students often find only two pairs of equal sides and stop, or they use the side without a reason.

Correct: Write the shared side as its own statement, AC‾≅AC‾\overline{AC} \cong \overline{AC}, with the reason reflexive property. It counts as the third pair.

Deciding congruence from equal perimeters

Why it's wrong: Triangles with sides 3, 4, 5 and 4, 4, 4 both have perimeter 12, but they have different shapes.

Correct: SSS needs three separate pairs of equal sides, not just the same total.

Interactive Visual

Congruent triangles

576ABC576DEF△ABC ≅ △DEF

SSSAll three pairs of sides are equal.

  • AB = DE = 5
  • BC = EF = 7
  • CA = FD = 6

Tap a criterion to see which matching parts it needs. Parts with the same marks are equal.

Function graph

-8-7-6-5-4-3-2-112345678-6-5-4-3-2-112345670xyA2√5B3√2C√26D2√5E3√2F√26

Interactive Sandbox

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Practice Problems

18 problems
Problem 1 of 18
Easy
Which information is enough to prove two triangles congruent by SSS?

Why It Matters

SSS is the criterion you can check with nothing but a ruler, and it explains why triangles are everywhere in building:
  • Bridges and roofs: a triangle made of three fixed beams cannot change shape, so trusses are built from triangles. A square frame with hinged corners collapses into a rhombus, a triangle never does
  • Manufacturing: to check that a triangular part matches the template, workers measure the three sides only
  • Maps and surveying: with coordinates you can prove two triangular plots are identical just by computing distances
  • Proofs: SSS is often the first step of a proof, and then CPCTC (Corresponding Parts of Congruent Triangles are Congruent) gives you equal angles for free
In the lessons on congruent triangles and on ASA and AAS you met the other criteria. SSS is the only one that needs no angle at all.

Real World Applications

Roof Trusses

Carpenters build roofs from identical triangles because a triangle with fixed sides cannot be pushed out of shape.

Example:

Every truss of a garden shed is made from beams of 3 m, 4 m and 5 m. By SSS all trusses are congruent, so the roof is the same along its whole length.

1Try It Yourself

A carpenter builds 12 identical roof trusses. Each truss is a triangle with beams of 3 m, 4 m and 5 m.

How many meters of beam does the carpenter need in total?

Step 1: Write the mathematical expression

Total beam length in meters:

Fields on a Map Grid

Planners use coordinates and the distance formula to compare plots of land without visiting them.

Example:

Field 1 has corners (0,0)(0, 0), (6,0)(6, 0) and (0,8)(0, 8), field 2 has corners (10,2)(10, 2), (10,8)(10, 8) and (18,2)(18, 2). Both have sides 6, 8 and 10 grid units, so they are congruent by SSS.

2Try It Yourself

On a map one grid unit is 10 m. Field 2 has corners (10,2)(10, 2), (10,8)(10, 8) and (18,2)(18, 2).

How long is the slanted border of field 2, from (10,8)(10, 8) to (18,2)(18, 2), in meters?

Step 1: Write the mathematical expression

Length in meters:

Quality Control of Tiles

A tile factory checks triangular tiles against a template by measuring the three sides.

Example:

The template has sides 12 cm, 15 cm and 20 cm. A tile with sides 20 cm, 12 cm and 15 cm passes: the order of measuring does not matter, only the three pairs.

3Try It Yourself

The template has sides 12 cm, 15 cm and 20 cm. A test tile has sides 15 cm, 20 cm and xx cm.

What must xx be for the tile to be congruent to the template?

Step 1: Write the mathematical expression

Value of x:

Key Takeaways

  • 1SSS: if three sides of one triangle are congruent to three sides of another, the triangles are congruent
  • 2No angles are needed: three side lengths fix a triangle completely (triangle rigidity)
  • 3Pair shortest with shortest, middle with middle, longest with longest, then write the vertices in matching order
  • 4On a coordinate grid, find each side with the distance formula d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
  • 5A side shared by two triangles is congruent to itself (reflexive property) and counts as one pair
  • 6After SSS, CPCTC gives the matching angles; SSS does not work for quadrilaterals

Frequently Asked Questions

Try to build a triangle from three sticks of fixed length. Once the ends are joined there is only one shape, you cannot open or close the angles. Every triangle with those three sides is the same triangle, maybe turned or flipped. That is why three sides are enough.
Try to build a triangle from three sticks of fixed length. Once the ends are joined there is only one shape, you cannot open or close the angles. Every triangle with those three sides is the same triangle, maybe turned or flipped. That is why three sides are enough.
Four sticks joined at the corners can still move: a square with sides 4 can be pushed into a rhombus with sides 4. The sides stay the same, the angles change. Adding one diagonal splits the shape into two triangles and makes it rigid.
Not for deciding congruence: sides 5, 6, 8 and 8, 5, 6 give congruent triangles. It does matter for the congruence statement: the vertex letters must be written in the matching order.
SSS congruence needs three pairs of equal sides, so the triangles have the same size. SSS similarity only needs the three ratios of corresponding sides to be equal, so one triangle can be a scaled copy of the other.

Glossary

SSS criterion
Two triangles are congruent if their three pairs of corresponding sides are congruent
Corresponding sides
Sides that match in a congruence; they are named by the same positions in the congruence statement
Triangle rigidity
A triangle with fixed side lengths cannot change its shape or angles
Reflexive property
Every segment is congruent to itself, used for a side shared by two triangles
CPCTC
Corresponding Parts of Congruent Triangles are Congruent: once two triangles are congruent, all matching sides and angles are equal
Distance formula
d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}, the length of the segment between two points

Formula Card

SSS Criterion

AB=DE, BC=EF, CA=FD⇒△ABC≅△DEFAB = DE,\ BC = EF,\ CA = FD \Rightarrow \triangle ABC \cong \triangle DEF

Three pairs of congruent sides prove the triangles congruent

Distance Formula

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Length of a side from the coordinates of its endpoints

Reflexive Property

AC‾≅AC‾\overline{AC} \cong \overline{AC}

A shared side is congruent to itself

CPCTC

△ABC≅△DEF⇒∠A≅∠D\triangle ABC \cong \triangle DEF \Rightarrow \angle A \cong \angle D

Corresponding parts of congruent triangles are congruent

Today's challengeOne problem, three tries

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Introduction to Similarity

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Congruent Triangles

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