Congruent Symbol (≅): Meaning, ≅ vs ≡ vs ≈, and How to Type It

The congruent symbol ≅ means "has the same size and shape": △ABC ≅ △DEF says that the two triangles match exactly, so every side and every angle of one equals the corresponding side and angle of the other. The sign combines a tilde (similar shape) with an equals sign (equal size). In LaTeX it is \cong, in HTML it is ≅, and in Word you can type 2245 followed by Alt + X.

This guide explains what congruence means in geometry and in modular arithmetic, how ≅ differs from the look-alike signs ≡, ≈, ∼, ≃, and ≍, and how to type each one on Windows, Mac, phones, Google Docs, HTML, and LaTeX.

Copy and Paste the Congruent Symbol and Its Relatives

• ≅ — approximately equal to, used as the congruent sign (U+2245)
• ≇ — neither approximately nor actually equal to, "not congruent" (U+2247)
• ≡ — identical to, used for congruence modulo n (U+2261)
• ≢ — not identical to (U+2262)
• ≈ — almost equal to (U+2248)
• ∼ — tilde operator, "similar to" (U+223C)
• ≃ — asymptotically equal to (U+2243)
• ≍ — equivalent to (U+224D)
• ≌ — all equal to (U+224C)
• △ — white up-pointing triangle, for naming triangles (U+25B3)

What the Congruent Symbol Means in Geometry

Two figures are congruent when one can be moved onto the other by a rigid motion: a translation, a rotation, a reflection, or any combination of the three. Nothing stretches or shrinks. For triangles this is the subject of the familiar congruence tests:

• SSS — three pairs of equal sides
• SAS — two sides and the angle between them
• ASA — two angles and the side between them
• AAS — two angles and a non-included side
• HL — the hypotenuse and one leg of a right triangle

The order of the letters matters. △ABC ≅ △DEF states that A corresponds to D, B to E, and C to F, so AB = DE, BC = EF, and ∠A = ∠D. Segments and angles are congruent too: AB ≅ CD means the segments have equal length, and ∠ABC ≅ ∠DEF means the angles have equal measure. Many textbooks reserve ≅ for figures and = for the numbers that measure them, so AB ≅ CD but AB = CD = 5 cm.

Congruent Versus Similar

Similar figures have the same shape but not necessarily the same size: one is a scaled copy of the other. The similarity sign is the plain tilde, △ABC ∼ △DEF, and every pair of congruent figures is also similar with a scale factor of 1. Reading the glyph tells you which is which: ∼ alone is shape, ≅ is shape plus size.

The Congruence Symbol in Number Theory

Modular arithmetic uses a different sign for a different kind of congruence. The statement a ≡ b (mod n), read "a is congruent to b modulo n," means that n divides a − b, or equivalently that a and b leave the same remainder when divided by n:

• 17 ≡ 2 (mod 5), because 17 − 2 = 15 is a multiple of 5
• 2024 ≡ 0 (mod 8), because 2024 is divisible by 8
• −1 ≡ 11 (mod 12), the fact that makes clock arithmetic work

The triple bar ≡ was introduced by Gauss in Disquisitiones Arithmeticae (1801), and he chose it precisely because congruence behaves like equality: you can add, subtract, and multiply congruences modulo the same n just as you would equations. The same ≡ sign also means "identically equal" or "defined as" in analysis and physics, where f(x) ≡ 0 says a function is zero for every x, not just at a particular point.

≅ vs ≡ vs ≈ vs ∼ vs ≃ vs ≍: A Comparison

Abstract algebra borrows ≅ for isomorphism: G ≅ H means two groups (or rings, vector spaces, graphs) have identical structure under a relabelling of their elements, which is the algebraic version of "same shape and size." Statistics borrows ∼ for "is distributed as," X ∼ N(0, 1). Analysis uses ≃ and ≍ for different flavours of "behaves the same for large inputs," and ≈ for a numerical approximation. Because the meanings overlap at the edges, papers often state in a notation section which sign stands for which relation.

For the symbols on either side of these, see https://www.useoctree.com/blog/not-equal-sign-meaning-how-to-type for ≠ and https://www.useoctree.com/blog/greater-than-less-than-symbols-guide for inequalities.

How to Type the Congruent Symbol

Windows: ≡ and ≈ have classic three-digit Alt codes (Alt + 240 and Alt + 247) because they sit in the old code page 437. ≅ does not, so in Word type the hex value 2245 and press Alt + X, or hold Alt and type the decimal 8773 on the numeric keypad; the long form works in Word and WordPad but not in every application. Insert → Symbol → More Symbols with the subset set to Mathematical Operators lists all of them, and ≇ (not congruent) sits two places after ≅.

Mac: Option + X gives ≈ on the US layout, but there is no Option shortcut for ≅ or ≡. Open the Character Viewer with Control + Command + Space, search "congruent" or "identical", and double-click the symbol. For frequent use, add a text replacement in System Settings → Keyboard so ;cong expands to ≅.

iPhone and Android: Neither stock keyboard includes ≅ or ≡. Copy the symbol from the list above, or create a text replacement (iOS: Settings → General → Keyboard → Text Replacement; Gboard: Settings → Dictionary → Personal dictionary).

Google Docs and Word: In Google Docs use Insert → Special characters and search "congruent", or draw the symbol in the search box. In Word, use Insert → Symbol, or the Alt + X trick above. Inside either app's equation editor, typing \cong followed by a space produces ≅, and \equiv produces ≡.

Congruent in LaTeX

\cong, \equiv, \approx, \sim, \simeq, and \asymp are all defined by the LaTeX kernel and are relation symbols, so they take correct spacing automatically. The not-congruent sign ≇ is \ncong from the amssymb package; the other negations are built with the \not prefix, so \not\equiv gives ≢ and \not\approx gives ≉ without any extra package. For modular congruences, \pmod{n} typesets the parenthesised "(mod n)" with the standard spacing, and \bmod gives a bare "mod" operator for expressions like a mod n.

1\documentclass{article}
2\usepackage{amsmath}
3\usepackage{amssymb}
4\begin{document}
5Geometry: $\triangle ABC \cong \triangle DEF$, $\overline{AB} \cong \overline{CD}$,
6$\angle A \cong \angle D$, and $\triangle ABC \sim \triangle DEF$ but
7$\triangle ABC \ncong \triangle PQR$.
8
9Number theory: $17 \equiv 2 \pmod{5}$, $a \not\equiv b \pmod{n}$,
10and $r = a \bmod n$.
11
12Algebra and analysis: $G \cong H$, $X \simeq Y$, $f \asymp g$, $\pi \approx 3.14$,
13and $X \sim \mathcal{N}(0, 1)$.
14\end{document}
Copy

All of these commands work only in math mode, so wrap them in $...$ inside a sentence. \triangle draws the △ used to name triangles, and \overline{AB} puts the bar over a segment name. The full list of relation symbols with code points and commands is at https://tools.useoctree.com/symbols/relations, and https://www.useoctree.com/learn/mathematical-expressions covers math mode basics.

Related Symbols

For ⊥ and ∥ see https://www.useoctree.com/blog/perpendicular-parallel-symbols-meaning-how-to-type.

FAQ

What does the ≅ symbol mean?

≅ means "is congruent to." In geometry it says two figures have exactly the same shape and size, so △ABC ≅ △DEF means the triangles can be laid on top of each other perfectly. In algebra the same sign means "is isomorphic to."

What is the difference between ≅ and ≡?

≅ is geometric congruence or isomorphism; ≡ is congruence modulo a number, as in 17 ≡ 2 (mod 5), or an identity that holds for every value of a variable. They are different Unicode characters (U+2245 and U+2261) with different LaTeX commands, \cong and \equiv.

What is the difference between ≅ and ≈?

≅ is an exact relationship between shapes or structures; ≈ means two numbers are approximately equal, as in √2 ≈ 1.414. Unicode confusingly names ≅ "approximately equal to," but in mathematics it is almost always read as congruent.

How do I type the congruent symbol on a keyboard?

In LaTeX write \cong in math mode. In Word type 2245 and press Alt + X, or hold Alt and type 8773 on the numeric keypad. On a Mac open the Character Viewer (Control + Command + Space) and search "congruent". In HTML use ≅.

How do I write "not congruent"?

The symbol is ≇ (U+2247). In LaTeX load amssymb and write \ncong; in HTML use ≇; in Word type 2247 and press Alt + X.

Writing Congruence Cleanly in Your Documents

Use \cong for figures and isomorphisms, \equiv with \pmod{n} for modular arithmetic, \sim for similarity, and \approx only for numbers. Octree, the AI LaTeX editor at https://www.useoctree.com, autocompletes each relation symbol and previews the compiled result instantly, so your proofs read exactly as intended.