Congruence theorem

In planar geometry, a congruence theorem is a statement that can be used to easily prove the congruence of triangles . Triangles are congruent (congruent) if they are the same in shape and area . The triangular congruence (i.e. the congruence of triangles) forms an equivalence relation , that is, congruent triangles can be regarded as equal.

Congruence clauses

In the usual designations of the four congruence sentences, "S" stands for a side length and "W" for an angle:

SSS sentence (first congruence sentence)

Two triangles that match in their three side lengths are congruent.

SWS sentence (second congruence sentence)

Two triangles that match in two side lengths and in the included angle are congruent.

WSW theorem (third congruence theorem)

Two triangles that coincide in one side and in the angles adjacent to this side are congruent.

In addition to the theorem of the sum of the interior angles in the triangle, this also includes the following theorem:

SWW set

Two triangles which coincide in one side length, an angle lying on this side and the angle opposite this side are congruent.

Note: However, two triangles that coincide in two angles and in one side length are not necessarily congruent, if it is not known which of the given angles lie on the given side. Using information on one side and two angles, three generally non-congruent triangles can be constructed, depending on whether the first, second or both angles are adjacent to the side.

SSW sentence (fourth congruence sentence)

Two triangles that coincide in two lengths and in the angle opposite the longer side are congruent.

The restriction to a SSW sentence that does not exist in general is expressed by a corresponding spelling or identification (e.g. SsW, Ssw or SSW _g , see the figure below ).

If two triangles match in two or all three interior angles , they are not necessarily congruent. However, they are similar .

The following figure shows the sizes in which two triangles must match for each of the four congruence sets.

From left to right: SSS, WSW, SWS, SSW.

proofs

Classically, one proves the congruence theorems by specifying constructions with compasses and ruler, which construct a second triangle from the corresponding given sizes. If this only works in one way, the two triangles are congruent. With designations as in the figure above, this goes as follows:

SSS

Given , and . Plot a stretch of length ; the circle with radius and the order of radius intersect at two points and , thereby, two mirror-symmetrical (that is congruent) triangles and yield. If you commit to an orientation, the triangle is even clear. This also applies to the following constructions: ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle BC}$ ${\displaystyle a}$ ${\displaystyle C}$ ${\displaystyle b}$ ${\displaystyle B}$ ${\displaystyle c}$ ${\displaystyle A_{1}}$ ${\displaystyle A_{2}}$ ${\displaystyle A_{1}BC}$ ${\displaystyle A_{2}BC}$

WSW

Given , and . Plot a stretch of length ; the half-line (the ray), which encloses with the angle , and the one, which encloses with the angle , intersect at a point . ${\displaystyle a}$ ${\displaystyle \beta }$ ${\displaystyle \gamma }$ ${\displaystyle BC}$ ${\displaystyle a}$ ${\displaystyle B}$ ${\displaystyle BC}$ ${\displaystyle \beta }$ ${\displaystyle C}$ ${\displaystyle BC}$ ${\displaystyle -\gamma }$ ${\displaystyle A}$

SWS

Given , and . Two half-lines (rays), associated with a vertex angle include wearing the length and from to and to find. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle \gamma }$ ${\displaystyle C}$ ${\displaystyle \gamma }$ ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle A}$ ${\displaystyle B}$

SSW congruence theorem

SSW

Given , and (where ). Construct two half-lines that include the angle as vertices ; mark the shorter distance on one leg to find; the circle around with radius intersects the other leg at one point . ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle \gamma }$ ${\displaystyle c>b}$ ${\displaystyle C}$ ${\displaystyle \gamma }$ ${\displaystyle b}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle c}$ ${\displaystyle B}$

The picture on the right shows that the angle on the SSW sentence must be opposite the longer side . Otherwise one would have triangles that match in three parts (SSW), but are not congruent: The two triangles and match in terms of side lengths and as well as in angle . The side lengths and differ, however. ${\displaystyle A_{1}BC}$ ${\displaystyle A_{2}BC}$ ${\displaystyle a}$ ${\displaystyle c}$ ${\displaystyle \gamma =\angle ACB}$ ${\displaystyle b_{1}}$ ${\displaystyle b_{2}}$

Remarks

• In Hilbert's system of axioms of Euclidean geometry , SWS has the rank of an axiom , the others are proven from this and the other axioms. Hilbert recognized this as necessary because Euclid's ideas of proof were used in the traditional structure that could not be derived purely logically from his axioms and postulates, but based on the vividly plausible free mobility of the triangles.

• Under certain circumstances it is also possible to construct a triangle from three other determinants, under which, for example, the incircle radius, the circumference radius, the area or the heights occur. However, the associated congruence statements are not counted among the classic congruence clauses.

• In the spherical geometry , the situation differs partially. So there two (spherical) triangles are already congruent and not just similar if they match in the three interior angles. The specification of the third angle is also no longer redundant ( spherical excess ).

Congruence Proofs

The four theorems of congruence form the basis of a proof procedure that is often used in elementary geometry: In a proof of congruence , one justifies the equality of two line lengths or two angular quantities by first showing the congruence of two suitable triangles and then deducing the equality of the corresponding side lengths or angles .

literature

• Hans Schupp: Elementary Geometry. UTB, Stuttgart 1977. ISBN 3-506-99189-2 , p. 76.

Individual evidence

1. ↑ Hartmut Wellstein: Elementary Geometry . Vieweg + Teubner, Wiesbaden 2009, ISBN 978-3-8348-0856-1 , pp. 12 .