» 
Arabic Bulgarian Chinese Croatian Czech Danish Dutch English Estonian Finnish French German Greek Hebrew Hindi Hungarian Icelandic Indonesian Italian Japanese Korean Latvian Lithuanian Malagasy Norwegian Persian Polish Portuguese Romanian Russian Serbian Slovak Slovenian Spanish Swedish Thai Turkish Vietnamese
Arabic Bulgarian Chinese Croatian Czech Danish Dutch English Estonian Finnish French German Greek Hebrew Hindi Hungarian Icelandic Indonesian Italian Japanese Korean Latvian Lithuanian Malagasy Norwegian Persian Polish Portuguese Romanian Russian Serbian Slovak Slovenian Spanish Swedish Thai Turkish Vietnamese

definition - Congruence (geometry)

definition of Wikipedia

   Advertizing ▼

Wikipedia

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 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 translations, rotations and reflections. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object.

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 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, 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.

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.[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

   
               

 

All translations of Congruence (geometry)


sensagent's content

  • definitions
  • synonyms
  • antonyms
  • encyclopedia

Dictionary and translator for handheld

⇨ New : sensagent is now available on your handheld

   Advertising ▼

sensagent's office

Shortkey or widget. Free.

Windows Shortkey: sensagent. Free.

Vista Widget : sensagent. Free.

Webmaster Solution

Alexandria

A windows (pop-into) of information (full-content of Sensagent) triggered by double-clicking any word on your webpage. Give contextual explanation and translation from your sites !

Try here  or   get the code

SensagentBox

With a SensagentBox, visitors to your site can access reliable information on over 5 million pages provided by Sensagent.com. Choose the design that fits your site.

Business solution

Improve your site content

Add new content to your site from Sensagent by XML.

Crawl products or adds

Get XML access to reach the best products.

Index images and define metadata

Get XML access to fix the meaning of your metadata.


Please, email us to describe your idea.

WordGame

The English word games are:
○   Anagrams
○   Wildcard, crossword
○   Lettris
○   Boggle.

Lettris

Lettris is a curious tetris-clone game where all the bricks have the same square shape but different content. Each square carries a letter. To make squares disappear and save space for other squares you have to assemble English words (left, right, up, down) from the falling squares.

boggle

Boggle gives you 3 minutes to find as many words (3 letters or more) as you can in a grid of 16 letters. You can also try the grid of 16 letters. Letters must be adjacent and longer words score better. See if you can get into the grid Hall of Fame !

English dictionary
Main references

Most English definitions are provided by WordNet .
English thesaurus is mainly derived from The Integral Dictionary (TID).
English Encyclopedia is licensed by Wikipedia (GNU).

Copyrights

The wordgames anagrams, crossword, Lettris and Boggle are provided by Memodata.
The web service Alexandria is granted from Memodata for the Ebay search.
The SensagentBox are offered by sensAgent.

Translation

Change the target language to find translations.
Tips: browse the semantic fields (see From ideas to words) in two languages to learn more.

last searches on the dictionary :

4140 online visitors

computed in 0.047s

I would like to report:
section :
a spelling or a grammatical mistake
an offensive content(racist, pornographic, injurious, etc.)
a copyright violation
an error
a missing statement
other
please precise:

Advertize

Partnership

Company informations

My account

login

registration

   Advertising ▼