Congruent: what it means in math

In math, congruent means exactly the same shape and size. Two figures are congruent if one can be moved onto the other by sliding, turning or flipping it, so that they match perfectly. The symbol for congruent is ≅, so △ABC ≅ △DEF says triangle ABC is congruent to triangle DEF. Outside math, congruent means in agreement or in harmony.

What does congruent mean?

Congruent figures are identical copies: every side and every angle of one matches a side and angle of the other. Their position can be different, and one can be a mirror image of the other, but nothing is stretched or shrunk.

Matching tick marks show sides of equal length: one tick on AB and DE, two on BC and EF, three on CA and FD. The triangles face opposite ways, but they are still congruent, because a reflection, a flip, puts one exactly on top of the other.

The congruent symbol

The symbol ≅ is an equals sign with a wavy line on top: the wave stands for "same shape" and the equals sign for "same size". Write it between two figures, segments or angles: AB ≅ DE means segment AB is congruent to segment DE, and ∠A ≅ ∠D means the angles have the same measure.

Congruent shapes and rigid motions

A rigid motion moves a figure without changing its size or shape. There are three kinds:

Two figures are congruent when one can be carried onto the other by a sequence of these moves. That is the definition used in US schools under the Common Core standards: grade 8 students describe the slides, turns and flips that match two congruent figures, and high school geometry builds the triangle rules from it. A dilation (enlargement) is not a rigid motion, because it changes the size.

Example 1: a translation

Triangle PQR has corners P(1, 1), Q(4, 1) and R(1, 3). Slide it 5 right and 2 up.

Add 5 to each x-coordinate and 2 to each y-coordinate: P′(6, 3), Q′(9, 3) and R′(6, 5). PQ and P′Q′ are both 3 long, PR and P′R′ are both 2, and QR and Q′R′ are both √13 ≈ 3.61. Every length is kept, so △PQR ≅ △P′Q′R′.

Congruent triangles

Two triangles are congruent when all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. That is six facts, but you never need to check all six. Five shortcuts each prove two triangles congruent from just three facts.

In each figure below, the marks show what has to match a second triangle. Ticks mean equal sides, and arcs mean equal angles.

SSS: side, side, side

If all three sides of one triangle equal the three sides of another, the triangles are congruent. Three side lengths fix a triangle completely, which is why triangles make rigid frames in bridges and roofs.

SAS: side, angle, side

Two sides and the angle between them, the included angle, match. The angle must be the one formed by the two sides.

ASA: angle, side, angle

Two angles and the side between them match.

AAS: angle, angle, side

Two angles and a side that is not between them match. This works because the angles in any triangle total 180°, so two matching angles make the third match too, and AAS turns into ASA.

HL: hypotenuse, leg

For right triangles only: if the hypotenuse and one leg match, the triangles are congruent. The Pythagorean theorem fixes the third side, so HL is really SSS in disguise.

Why SSA and AAA don't prove congruence

Not every set of three matching facts works. AAA fails because two triangles can have the same three angles but different sizes: those triangles are similar, not congruent. SSA, two sides and an angle that is not between them, fails because the same measurements can make two different triangles:

Swing a side of 4 from B and it meets the other arm of the 30° angle in two places, giving triangle ABC1 and triangle ABC2. Same angle, same two sides, different triangles. (The one exception is a right angle, which is why HL works.)

Check you can pick the right rule.

CPCTC: corresponding parts

Once you have proved two triangles congruent, every other pair of matching parts is congruent too. Geometry proofs give this a name: CPCTC, short for "corresponding parts of congruent triangles are congruent". It is how you prove that two segments or angles are equal: show they belong to congruent triangles first.

Example 2: a short proof

In triangle ABC, AB ≅ CB, and the segment BD cuts angle ABC into two equal angles, with D on AC. Prove that △ABD ≅ △CBD, and so AD ≅ CD.

Using congruence to find missing values

Example 3: a missing side

△ABC ≅ △DEF, AB = 3x − 4 and DE = 2x + 5. Find x and the length of AB.

AB and DE are corresponding sides, so they are equal:

So AB = 3(9) − 4 = 23, and DE = 2(9) + 5 = 23 as well.

Example 4: a missing angle

△PQR ≅ △XYZ, ∠P = 48° and ∠Q = 75°. Find ∠Z.

∠Z corresponds to ∠R, the third angle of the first triangle: 180° − 48° − 75° = 57°. So ∠Z = 57°.

Congruent vs similar

Every pair of congruent figures is also similar, with a scale factor of 1. Most similar figures are not congruent.

Congruent angles and segments

• Congruent segments have the same length, shown with matching tick marks.
• Congruent angles have the same measure, shown with matching arcs. Vertical angles are always congruent, and so are the base angles of an isosceles triangle.
• Congruent circles have the same radius.

Other meanings of congruent

In everyday English, congruent means in agreement or consistent with something: "her actions were congruent with her values". The word comes from the Latin congruere, to meet together or agree.

In number theory, two whole numbers are congruent modulo n when they leave the same remainder after dividing by n. For example, 17 and 29 are congruent modulo 12, written 17 ≡ 29 (mod 12), because both leave a remainder of 5. Clock arithmetic works this way: 29 hours after midnight, a 12-hour clock reads the same as 17 hours after.

Common mistakes

• Writing the letters in the wrong order. △ABC ≅ △EDF and △ABC ≅ △DEF say different things. Match the corners.
• Using SSA or AAA. Neither proves congruence.
• Using a non-included angle for SAS. The angle must be between the two sides.
• Thinking a flipped figure is not congruent. Reflections keep size and shape, so mirror images are congruent.
• Confusing congruent with equal. Lengths and angle measures are equal (AB = DE). The figures themselves are congruent (AB ≅ DE).
• Calling different-sized similar figures congruent. Same shape is not enough: the size must match too.

Practice with a short timed paper

Congruence, transformations and triangle proofs are central to high school geometry. The daily Geometry, Algebra 2 & SAT Math paper is a short timed practice test of 8 questions, with a new one every day. For the angle facts behind these proofs, see right angles, complementary angles and scalene triangles.

Read next

Maths · GeometryLine segment: definition, notation, length and midpointWhat a line segment is, how it differs from a line and a ray, how to name it, and how to find its length and midpoint on a number line or a graph.