Congruence (geometry)


Congruence (geometry)
An example of congruence. The two figures on the left are congruent, while the third is similar to them. The last figure is neither similar nor congruent to any of the others. Note that congruences alter some properties, such as location and orientation, but leave others unchanged, like distance and angles. The unchanged properties are called invariants.

In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object. 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 translations, rotations and reflections.

The related concept of similarity applies if the objects differ in size but not in shape.

Contents

Definition of congruence in analytic geometry

In a Euclidean system, congruence is fundamental; it is the counterpart of equality for numbers. In analytic geometry, congruence may be defined intuitively thus: two mappings of figures onto one Cartesian coordinate system are congruent if and only if, for any two points in the first mapping, the Euclidean distance between them is equal to the Euclidean distance between the corresponding points in the second mapping.

A more formal definition: two subsets A and B of Euclidean space Rn are called congruent if there exists an isometry f : RnRn (an element of the Euclidean group E(n)) with f(A) = B. Congruence is an equivalence relation.

Congruence of triangles

See also Solution of triangles.

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

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.

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.

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. (In British usage, ASA and AAS are usually combined into a single condition AAcorrS - any two angles and a corresponding side.)[1]
  • RHS (Right-angle-Hypotenuse-Side): 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 four possible cases:

If two triangles satisfy the SSA condition and the corresponding angles are either obtuse or right, then the two triangles are congruent. In this situation, the length of the side opposite the angle will be greater than the length of the adjacent side. 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 Pythagoras' 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 greater than or equal to the length of the adjacent side, 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 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.

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.[2]

See also

References

  1. ^ Parr, H. E. (1970). Revision Course in School mathematics. Mathematics Textbooks Second Edition. G Bell and Sons Ltd.. ISBN 0-7135-1717-4. 
  2. ^ Cornel, Antonio (2002). Geometry for Secondary Schools. Mathematics Textbooks Second Edition. Bookmark Inc.. ISBN 971-569-441-1. 

External links


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Congruence — is the state achieved by coming together, the state of agreement. The Latin congruō meaning “I meet together, I agree”. As an abstract term, congruence means similarity between objects. Congruence, as opposed to equivalence or approximation, is a …   Wikipedia

  • Congruence relation — See congruence (geometry) for the term as used in elementary geometry. In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is… …   Wikipedia

  • Congruence of triangles — The congruence of triangles may refer to: Congruence (geometry)#Congruence of triangles Solution of triangles This disambiguation page lists articles associated with the same title. If an internal link led you here, you may wish …   Wikipedia

  • Congruence (manifolds) — In the theory of smooth manifolds, a congruence is the set of integral curves defined by a nonvanishing vector field defined on the manifold. Congruences are an important concept in general relativity, and are also important in parts of… …   Wikipedia

  • Congruence (general relativity) — In general relativity, a congruence (more properly, a congruence of curves) is the set of integral curves of a (nowhere vanishing) vector field in a four dimensional Lorentzian manifold which is interpreted physically as a model of spacetime.… …   Wikipedia

  • geometry — /jee om i tree/, n. 1. the branch of mathematics that deals with the deduction of the properties, measurement, and relationships of points, lines, angles, and figures in space from their defining conditions by means of certain assumed properties… …   Universalium

  • congruence — congruent ► ADJECTIVE 1) in agreement or harmony. 2) Geometry (of figures) identical in form. DERIVATIVES congruence noun. ORIGIN from Latin congruere agree, meet together …   English terms dictionary

  • congruence theorem — formula which declares triangles to be exactly coinciding (Geometry) …   English contemporary dictionary

  • congruence — /ˈkɒŋgruəns/ (say konggroohuhns) noun 1. the fact or condition of agreeing; agreement. 2. Geometry the state of being congruent (def. 2). 3. Mathematics the property of being congruent (def. 3). Also, congruency …   Australian English dictionary

  • Similarity (geometry) — Geometry = Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. One can be obtained from the other by uniformly stretching , possibly with additional rotation,… …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.