The congruent definition in geometry is simple: two figures are congruent when they have exactly the same shape and the same size, so one could sit perfectly on top of the other. Outside maths, congruent means in agreement or consistent, as when your actions are congruent with your words.
Both senses share one picture: things that fit together without any gap or clash. A student meets the word in a geometry lesson; a manager meets it in a report saying a new policy is congruent with company values. The idea of a clean match is the same.
Entry at a glance
Meaning
Same shape and size; or consistent and in harmonyPart of speech
AdjectivePronunciation
KONG-groo-uhnt; also kuhn-GROO-uhntWord forms
congruence or congruency (noun), congruently (adverb), incongruent (opposite)Symbol
≅ in geometry, as in triangle ABC ≅ triangle DEFOrigin
Latin congruens, from congruere, to agree, come together; English from the early 1400sRegister
Technical in maths; formal in general use
Congruent definition in three senses
- Geometry: identical in shape and size. Two squares with 5 cm sides are congruent. Position and direction do not matter, so a shape that has been turned, slid or flipped is still congruent to the original.
- Number theory: two whole numbers are congruent modulo n when they leave the same remainder after division by n. For example, 17 and 5 are congruent modulo 12, which is why 17:00 on a 24-hour clock is 5 o'clock.
- General use: in agreement, suitable or consistent. "Her calm voice was congruent with the message of the speech."
If you must define congruent just once, the definition of congruent that covers all three is this: things that line up exactly according to the rule being used.
Congruent triangles: the tests
You do not need to measure every side and angle to prove two triangles congruent. School geometry uses short tests:
| Rule | What must match |
|---|---|
| SSS | All three sides |
| SAS | Two sides and the angle between them |
| ASA | Two angles and the side between them |
| AAS | Two angles and a side not between them |
| RHS (or HL) | In right-angled triangles: hypotenuse and one other side |
Matching three angles alone (AAA) is not enough. It shows the triangles are similar, the same shape, but one could be larger. Teachers often pair this topic with perpendicular lines and right angles.
Origin of the word
Congruent reached English during the 1400s, taken from Latin congruens, the present participle of congruere, meaning to agree or correspond. The literal picture is coming together: com- means together, joined to an old verb root meaning to fall or rush. The first English sense was suitable or harmoniously joined. The geometry sense, figures that can be laid exactly over one another, first appears in English writing around 1706.
Sentences with congruent
- Fold the paper along the line and check whether the two halves are congruent.
- The tiles on the kitchen floor are congruent hexagons, so they fit with no gaps.
- His hiring choices were not congruent with his talk about giving young people a chance.
- A good counsellor tries to keep body language congruent with what they say.
- In clock arithmetic, 26 is congruent to 2 modulo 12.
Congruent, similar, equal: the differences
Students often mix these up:
- Congruent: matching outline and matching measurements.
- Similar: same shape, size may differ (a photo and its enlargement).
- Equal: used for numbers and measurements, such as equal lengths or equal areas. Two shapes can have equal area without being congruent, like a 2 by 8 rectangle and a 4 by 4 square.
In general use, the opposite of congruent is incongruent or incongruous, meaning out of place or not fitting, such as a formal suit at a beach party. Being congruent with your values overlaps with integrity.
Grammar notes
Congruent usually takes with in general use (congruent with our goals) and to in geometry (triangle PQR is congruent to triangle XYZ). The noun is congruence; congruency is a less common variant that many textbooks also accept. For other terms from maths and science, see our science words page, or read about authentic for the idea of words matching actions.
Frequently asked questions
What does congruent mean in math?
It means two figures have exactly the same shape and size. You could cut one out and lay it perfectly over the other, even if you first had to rotate or flip it.
Are mirror images congruent?
Yes. A reflection does not change size or shape, so a shape and its mirror image are congruent.
What is the difference between congruent and similar?
Congruent shapes match in both shape and size. Similar shapes match in shape only, so one can be a scaled-up or scaled-down copy of the other.
Can congruent describe people or ideas?
Yes, in formal writing. You might say a plan is congruent with the budget, or that someone's behaviour is congruent with their beliefs, meaning they fit together.
How do you pronounce congruent?
The most common pronunciation is KONG-groo-uhnt. Some speakers stress the middle: kuhn-GROO-uhnt. Both are heard in classrooms.
What is the symbol for congruent?
Geometry uses ≅, an equals sign with a small wave above it. In number theory, ≡ (three lines) shows congruence modulo a number.