Triangle Congruence

An example of congruence. The two triangles on the left are congruent, while the third is similar to them. The last triangle is neither congruent nor similar to any of the others. Congruence permits alteration of some properties, such as location and orientation, but leaves others unchanged, like distances and angles. The unchanged properties are called invariants.

This diagram illustrates the geometric principle of angle-angle-side triangle congruence: given triangle ABC and triangle A'B'C', triangle ABC is congruent with triangle A'B'C' if and only if: angle CAB is congruent with angle C'A'B', and angle ABC is congruent with angle A'B'C', and BC is congruent with B'C'. Note hatch marks are used here to show angle and side equalities.

In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.

More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of rigid motions, namely a translation, a rotation, and a reflection. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. So two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted.

This diagram illustrates the geometric principle of angle-angle-side triangle congruence: given triangle ABC and triangle A'B'C', triangle ABC is congruent with triangle A'B'C' if and only if: angle CAB is congruent with angle C'A'B', and angle ABC is congruent with angle A'B'C', and BC is congruent with B'C'. Note hatch marks are used here to show angle and side equalities. In elementary geometry the word congruent is often used as follows. The word equal is often used in place of congruent for these objects.

• Two line segments are congruent if they have the same length.
• Two angles are congruent if they have the same measure.
• Two circles are congruent if they have the same diameter.

In this sense, two plane figures are congruent implies that their corresponding characteristics are "congruent" or "equal" including not just their corresponding sides and angles, but also their corresponding diagonals, perimeters, and areas.

The related concept of similarity applies if the objects have the same shape but do not necessarily have the same size. (Most definitions consider congruence to be a form of similarity, although a minority require that the objects have different sizes in order to qualify as similar.)

Congruence of triangles

The shape of a triangle is determined up to congruence by specifying two sides and the angle between them (SAS), two angles and the side between them (ASA) or two angles and a corresponding adjacent side (AAS). Specifying two sides and an adjacent angle (SSA), however, can yield two distinct possible triangles.

Two triangles are congruent if their corresponding sides are equal in length, and their corresponding angles are equal in measure.

If triangle ABC is congruent to triangle DEF, the relationship can be written mathematically as:

$\triangle \mathrm {ABC} \cong \triangle \mathrm {DEF}$.

In many cases it is sufficient to establish the equality of three corresponding parts and use one of the following results to deduce the congruence of the two triangles.

Determining congruence

Sufficient evidence for congruence between two triangles in Euclidean space can be shown through the following comparisons:

• SAS (side-angle-side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.
• SSS (side-side-side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent.
• ASA (angle-side-angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.

The ASA postulate was contributed by Thales of Miletus (Greek). In most systems of axioms, the three criteria – SAS, SSS and ASA – are established as theorems. In the School Mathematics Study Group system SAS is taken as one (#15) of 22 postulates.

• AAS (angle-angle-side): If two pairs of angles of two triangles are equal in measurement, and a pair of corresponding non-included sides are equal in length, then the triangles are congruent. AAS is equivalent to an ASA condition, by the fact that if any two angles are given, so is the third angle, since their sum should be 180°. ASA and AAS are sometimes combined into a single condition, AAcorrS – any two angles and a corresponding side.
• RHS (right-angle-hypotenuse-side), also known as HL (hypotenuse-leg): If two right-angled triangles have their hypotenuses equal in length, and a pair of shorter sides are equal in length, then the triangles are congruent.

Side-side-angle

The SSA condition (side-side-angle) which specifies two sides and a non-included angle (also known as ASS, or angle-side-side) does not by itself prove congruence. In order to show congruence, additional information is required such as the measure of the corresponding angles and in some cases the lengths of the two pairs of corresponding sides. There are a few possible cases:

If two triangles satisfy the SSA condition and the length of the side opposite the angle is greater than or equal to the length of the adjacent side (SSA, or long side-short side-angle), then the two triangles are congruent. The opposite side is sometimes longer when the corresponding angles are acute, but it is always longer when the corresponding angles are right or obtuse. Where the angle is a right angle, also known as the hypotenuse-leg (HL) postulate or the right-angle-hypotenuse-side (RHS) condition, the third side can be calculated using the Pythagorean theorem thus allowing the SSS postulate to be applied.

If two triangles satisfy the SSA condition and the corresponding angles are acute and the length of the side opposite the angle is equal to the length of the adjacent side multiplied by the sine of the angle, then the two triangles are congruent.

If two triangles satisfy the SSA condition and the corresponding angles are acute and the length of the side opposite the angle is greater than the length of the adjacent side multiplied by the sine of the angle (but less than the length of the adjacent side), then the two triangles cannot be shown to be congruent. This is the ambiguous case and two different triangles can be formed from the given information, but further information distinguishing them can lead to a proof of congruence.

Angle-angle-angle

In Euclidean geometry, AAA (angle-angle-angle) (or just AA, since in Euclidean geometry the angles of a triangle add up to 180°) does not provide information regarding the size of the two triangles and hence proves only similarity and not congruence in Euclidean space.

However, in spherical geometry and hyperbolic geometry (where the sum of the angles of a triangle varies with size) AAA is sufficient for congruence on a given curvature of surface.

CPCTC

This acronym stands for Corresponding Parts of Congruent Triangles are Congruent, which is an abbreviated version of the definition of congruent triangles.

In more detail, it is a succinct way to say that if triangles ABC and DEF are congruent, that is,

$\triangle ABC\cong \triangle DEF$,

with corresponding pairs of angles at vertices A and D; B and E; and C and F, and with corresponding pairs of sides AB and DE; BC and EF; and CA and FD, then the following statements are true:

${\overline {AB}}\cong {\overline {DE}}$

${\overline {BC}}\cong {\overline {EF}}$

${\overline {AC}}\cong {\overline {DF}}$

$\angle BAC\cong \angle EDF$

$\angle ABC\cong \angle DEF$

$\angle BCA\cong \angle EFD$.

The statement is often used as a justification in elementary geometry proofs when a conclusion of the congruence of parts of two triangles is needed after the congruence of the triangles has been established. For example, if two triangles have been shown to be congruent by the SSS criteria and a statement that corresponding angles are congruent is needed in a proof, then CPCTC may be used as a justification of this statement.

A related theorem is CPCFC, in which "triangles" is replaced with "figures" so that the theorem applies to any pair of polygons or polyhedrons that are congruent.

Notation

A symbol commonly used for congruence is an equals symbol with a tilde above it, ≅, corresponding to the Unicode character 'approximately equal to' (U+2245). In the UK, the three-bar equal sign ≡ (U+2261) is sometimes used.