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| | Nothing to tell about me really.<br>Nice to be here and a member of wmflabs.org.<br>I just wish I'm useful at all<br><br>My weblog [http://www.homeimprovementdaily.com homes ideas] |
| [[File:Congruent_non-congruent_triangles.svg|thumb|370px|An example of congruence. The two triangles on the left are congruent, while the third is [[Similarity (geometry)|similar]] to them. The last triangle is neither similar nor congruent to any of the others. Note that congruence permits alteration of some properties, such as location and orientation, but leaves others unchanged, like [[distance]] and [[angle]]s. The unchanged properties are called [[invariant (mathematics)|invariant]]s.]]
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| 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.<ref>{{cite web | url=http://web.cortland.edu/matresearch/OxfordDictionaryMathematics.pdf |title=Oxford Concise Dictionary of Mathematics, Congruent Figures | first1=C.|last1=Clapham|first2=J.|last2=Nicholson | publisher =Addison-Wesley | year =2009|page=167|accessdate=September 2013}}</ref> More formally, two sets of [[point (geometry)|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 [[translation (geometry)|translation]]s, [[rotation]]s and [[reflection (mathematics)|reflection]]s. 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.
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| [[File:congruence_en.gif|right|frame|Congruence of two triangles]]
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| In elementary geometry the word ''congruent'' is often used as follows.<ref>{{cite web|url=http://mathopenref.com/congruent.html|title=Congruence|publisher=Math Open Reference|year=2009|accessdate=September 2013}}</ref> The word ''equal'' is often used in place of ''congruent'' for these objects.
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| *Two [[line segment]]s are congruent if they have the same length.
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| *Two [[angle]]s are congruent if they have the same measure.
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| *Two [[circle]]s are congruent if they have the same diameter.
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| 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.
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| The related concept of [[Similarity (geometry)|similarity]] applies if the objects differ in size but not in shape.
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| ==Determining congruence of polygons==
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| [[File:quadrilateral_congruence.png|thumb|300px|The orange and green quadrilaterals are congruent; the blue is not congruent to them. All three have the same [[perimeter]] and [[area]]. (The ordering of the sides of the blue quadrilateral is "mixed" which results in two of the interior angles and one of the diagonals not being congruent.)]]
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| For two polygons to be congruent, they must have an equal number of sides (and hence an equal number—the same number—of vertices). Two polygons with ''n'' sides are congruent if they each have numerically identical sequences (even if clockwise for one polygon and counterclockwise for the other) side-angle-side-angle-... for ''n'' sides and ''n'' angles.
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| Congruence of polygons can be established graphically as follows:
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| *First, match and label the corresponding vertices of the two figures.
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| *Second, draw a vector from one of the vertices of the one of the figures to the corresponding vertex of the other figure. ''Translate'' the first figure by this vector so that these two vertices match.
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| *Third, ''rotate'' the translated figure about the matched vertex until one pair of corresponding sides matches.
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| *Fourth, ''reflect'' the rotated figure about this matched side until the figures match.
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| If at anytime the step cannot be completed, the polygons are not congruent.
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| ==Congruence of triangles==<!-- This section is linked from [[SAS]] and from [[congruence of triangles]], a redirect page. -->
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| : ''See also [[Solution of triangles]].''
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| Two [[triangle]]s are congruent if their corresponding [[Edge (geometry)|sides]] are equal in length and their corresponding [[angle]]s are equal in size.
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| If triangle ABC is congruent to triangle DEF, the relationship can be written mathematically as:
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| :<math>\triangle \mathrm{ABC} \cong \triangle \mathrm{DEF}</math>
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| 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.
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| [[Image:Congruent triangles.svg|thumb|200px|right|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.]]
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| ===Determining congruence===
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| Sufficient evidence for congruence between two triangles in [[Euclidean space]] can be shown through the following comparisons:
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| *'''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.
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| *'''SSS''' (Side-Side-Side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent.
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| *'''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. <br>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 [[theorem]]s. In the [[School Mathematics Study Group]] system '''SAS''' is taken as one (#15) of 22 postulates.
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| *'''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.)''<ref>{{cite book
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| | last = Parr
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| | first = H. E.
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| | title = Revision Course in School mathematics
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| | publisher = G Bell and Sons Ltd.
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| | series = Mathematics Textbooks Second Edition
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| | year = 1970
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| | isbn = 0-7135-1717-4}}</ref>
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| *'''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.
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| ====Side-side-angle====
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| 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:
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| 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, 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 [[Pythagoras' Theorem]] thus allowing the SSS postulate to be applied.
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| 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.
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| 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.
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| ====Angle-angle-angle====
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| 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 (geometry)|similarity]] and not congruence in Euclidean space.
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| 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.<ref>{{cite book
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| | last = Cornel
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| | first = Antonio
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| | authorlink = Antonio Coronel
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| | title = Geometry for Secondary Schools
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| | publisher = Bookmark Inc.
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| | series = Mathematics Textbooks Second Edition
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| | year = 2002
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| | doi =
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| | isbn = 971-569-441-1}}</ref>
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| ==Congruent triangles on a sphere==
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| {{Main|Solving triangles#Solving spherical triangles|Spherical trigonometry#Solution of triangles}}
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| As with plane triangles, on a sphere two triangles sharing the same sequence of angle-side-angle (ASA) are necessarily congruent (that is, they have three identical sides and three identical angles).<ref name=Bolin>Michael Bolin, "Exploration of Spherical Geometry", September 9, 2003, pp.6-7. http://math.iit.edu/~mccomic/420/notes/Bolin_spherical.pdf</ref> This can be seen as follows: One can situate one of the vertices with a given angle at the south pole and run the side with given length up the prime meridian. Knowing both angles at either end of the segment of fixed length ensures that the other two sides emanate with a uniquely determined trajectory, and thus will meet each other at a uniquely determined point; thus ASA is valid.
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| The congruence theorems side-angle-side (SAS) and side-side-side (SSS) also hold on a sphere; in addition, if two spherical triangles have an identical angle-angle-angle (AAA) sequence, they are congruent (unlike for plane triangles).<ref name=Bolin/>
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| The plane-triangle congruence theorem angle-angle-side (AAS) does not hold for spherical triangles.<ref>Hollyer, L., http://www.uh.edu/~hollyer/Module6/m6ppt/sld089.htm</ref> As in plane geometry, side-side-angle (SSA) does not imply congruence.
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| ==Definition of congruence in analytic geometry==
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| In a [[Euclidean geometry|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.
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| A more formal definition: two [[subset]]s ''A'' and ''B'' of [[Euclidean space]] '''R'''<sup>''n''</sup> are called congruent if there exists an [[isometry]] ''f'' : '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup> (an element of the [[Euclidean group]] ''E''(''n'')) with ''f''(''A'') = ''B''. Congruence is an [[equivalence relation]].
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| ==Congruent conic sections==
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| Two conic sections are congruent if their [[Eccentricity (mathematics)|eccentricities]] and one other distinct parameter characterizing them are equal. Their eccentricities establish their shapes, equality of which is sufficient to establish similarity, and the second parameter then establishes size. Since two [[circle]]s, [[parabola]]s, or [[rectangular hyperbola]]s always have the same eccentricity (specifically 0 in the case of circles, 1 in the case of parabolas, and <math>\sqrt{2}</math> in the case of rectangular hyperbolas), two circles, parabolas, or rectangular hyperbolas need to have only one other common parameter value, establishing their size, for them to be congruent.
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| ==Congruent polyhedra==
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| For two [[polyhedra]] with the same number ''E'' of edges, the same number of [[face (geometry)|faces]], and the same number of sides on corresponding faces, there exists a set of at most ''E'' measurements that can establish whether or not the polyhedra are congruent.<ref>Alexander Borisov, Mark Dickinson, and Stuart Hastings, "A congruence problem for polyhedra", ''[[American Mathematical Monthly]]'' 117, March 2010, pp. 232-249. [http://www.maa.org/programs/maa-awards/writing-awards/a-congruence-problem-for-polyhedra]</ref><ref>Alexa Creech, "A congruence problem" http://146.163.152.131/teaching/projects/creech_final.pdf</ref> For [[cube]]s, which have 12 edges, only 9 measurements are necessary.
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| ==See also==
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| *[[CPCTC]] (Corresponding parts of congruent triangles are congruent)
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| *[[Euclidean plane isometry]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| {{Commons category|Congruence}}
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| * [http://www.cut-the-knot.org/pythagoras/SSS.shtml The SSS] at [[Cut-the-Knot]]
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| * [http://www.cut-the-knot.org/pythagoras/SSA.shtml The SSA] at [[Cut-the-Knot]]
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| * Interactive animations demonstrating [http://www.mathopenref.com/congruentpolygons.html Congruent polygons], [http://www.mathopenref.com/congruentangles.html Congruent angles], [http://www.mathopenref.com/congruentlines.html Congruent line segments], [http://www.mathopenref.com/congruenttriangles.html Congruent triangles] at Math Open Reference
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| [[Category:Euclidean geometry]]
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| [[Category:Polygons]]
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| [[Category:Triangles]]
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