In geometry, two figures or objects are congruent


  • In geometry, two figures or protests are harmonious on the off chance that they have similar shape and estimate, or on the off chance that one has an indistinguishable shape and size from the reflect picture of the other.[1] 

  • All the more formally, two arrangements of focuses are called compatible if, and just on the off chance that, one can be changed into the other by an isometry, i.e., a mix of inflexible movements, in particular an interpretation, a revolution, and a reflection. This implies either protest can be repositioned and reflected (yet not resized) in order to harmonize accurately with the other question. So two particular plane figures on a bit of paper are compatible on the off chance that we can remove them and afterward coordinate them up totally. Turning the paper over is allowed. 

  • In basic geometry the word harmonious is frequently utilized as follows.[2] The word equivalent is regularly utilized as a part of place of compatible for these articles. 

  • Two line portions are harmonious in the event that they have similar length. 

  • Two points are consistent on the off chance that they have similar measure. 

  • Two circles are consistent on the off chance that they have similar distance across. 

  • In this sense, two plane figures are harmonious infers that their relating qualities are "consistent" or "rise to" including their comparing sides and points, as well as their comparing diagonals, borders and ranges. 

  • The related idea of similitude applies if the articles vary in size however not fit as a fiddle. 

  • Substance [hide] 

  • 1 Determining consistency of polygons 

  • 2 Congruence of triangles 

  • 2.1 Determining consistency 

  • 2.1.1 Side-side-point 

  • 2.1.2 Angle-point edge 

  • 3 Definition of consistency in logical geometry 

  • 4 Congruent conic areas 

  • 5 Congruent polyhedra 

  • 6 Congruent triangles on a circle 

  • 7 See moreover 

  • 8 References 

  • 9 External connections 

  • Deciding consistency of polygons 

  • The orange and green quadrilaterals are consistent; the blue is not compatible to them. Each of the three have similar border and range. (The requesting of the sides of the blue quadrilateral is "blended" which brings about two of the inside edges and one of the diagonals not being harmonious.) 

  • For two polygons to be consistent, they should have an equivalent number of sides (and henceforth an equivalent number—similar number—of vertices). Two polygons with n sides are compatible if and just on the off chance that they each have numerically indistinguishable successions (regardless of the possibility that clockwise for one polygon and counterclockwise for the other) side-edge side-point ... for n sides and n points. 

  • Compatibility of polygons can be set up graphically as takes after: 

  • To start with, match and name the comparing vertices of the two figures. 

  • Second, draw a vector from one of the vertices of the one of the figures to the relating vertex of the other figure. Decipher the primary figure by this vector so that these two vertices coordinate. 

  • Third, pivot the deciphered figure about the coordinated vertex until one sets of relating sides matches. 

  • Fourth, mirror the pivoted figure about this coordinated side until the figures coordinate. 

  • On the off chance that whenever the progression can't be finished, the polygons are not congruent.Two triangles are consistent if their relating sides are equivalent long, in which case their comparing edges are equivalent in size. 

  • In the event that triangle ABC is consistent to triangle DEF, the relationship can be composed scientifically as: 

  • {\displaystyle \triangle \mathrm {ABC} \cong \triangle \mathrm {DEF} } \triangle \mathrm {ABC} \cong \triangle \mathrm {DEF} 

  • Much of the time it is adequate to set up the uniformity of three relating parts and utilize one of the accompanying results to derive the harmoniousness of the two triangles. 

  • The state of a triangle is resolved up to harmoniousness by indicating two sides and the point between them (SAS), two edges and the side between them (ASA) or two edges and a comparing nearby side (AAS). Determining two sides and a neighboring edge (SSA), be that as it may, can yield two unmistakable conceivable triangles. 

  • Deciding harmoniousness 

  • Adequate proof for coinciding between two triangles in Euclidean space can be appeared through the accompanying correlations: 

  • SAS (Side-Edge Side): If two sets of sides of two triangles are equivalent long, and the included edges are equivalent in estimation, then the triangles are harmonious. 

  • SSS (Side-Side-Side): If three sets of sides of two triangles are equivalent long, then the triangles are compatible. 

  • ASA (Edge Side-Edge): If two sets of edges of two triangles are equivalent in estimation, and the included sides are equivalent long, then the triangles are harmonious. 

  • The ASA Hypothesize was contributed by Thales of Miletus (Greek). In many frameworks of sayings, the three criteria—SAS, SSS and ASA—are built up as hypotheses. In the School Science Concentrate on Gathering framework SAS is taken as one (#15) of 22 hypothesizes. 

  • AAS (Point Edge Side): If two sets of edges of two triangles are equivalent in estimation, and a couple of comparing non-included sides are equivalent long, then the triangles are compatible. (In English utilization, ASA and AAS are typically joined into a solitary condition AAcorrS - any two edges and a comparing side.)[3] For American use, AAS is proportional to an ASA condition, by the way that if any two edges are given, so is the third point, since their total ought to be 180°. 

  • RHS (Right-edge Hypotenuse-Side), otherwise called HL (Hypotenuse-Leg): If two right-calculated triangles have their hypotenuses level with long, and a couple of shorter sides are equivalent long, then the triangles are compatible. 

  • Side-side-edge 

  • The SSA condition (Side-Side-Edge) which indicates two sides and a non-included edge (otherwise called ASS, or Point Side-Side) does not independent from anyone else demonstrate coinciding. So as to show consistency, extra data is required, for example, the measure of the relating points and now and again the lengths of the two sets of comparing sides. There are a couple of conceivable cases: 

  • In the event that two triangles fulfill the SSA condition and the length of the side inverse the edge is more prominent than or equivalent to the length of the nearby side (SsA, or long side-short side-edge), then the two triangles are harmonious. The inverse side is once in a while longer when the comparing edges are intense, however it is constantly longer when the relating edges are correct or heartless. Where the point is a right edge, otherwise called the Hypotenuse-Leg (HL) hypothesize or the Right-edge Hypotenuse-Side (RHS) condition, the third side can be figured utilizing the Pythagorean Hypothesis accordingly permitting the SSS propose to be connected. 

  • In the event that two triangles fulfill the SSA condition and the relating edges are intense and the length of the side inverse the point is equivalent to the length of the neighboring side increased by the sine of the edge, then the two triangles are harmonious. 

  • On the off chance that two triangles fulfill the SSA condition and the comparing points are intense and the length of the side inverse the edge is more prominent than the length of the nearby side increased by the sine of the edge (however not exactly the length of the neighboring side), then the two triangles can't be appeared to be compatible. This is the vague case and two distinct triangles can be framed from the given data, yet additional data recognizing them can prompt a proof of harmoniousness. 

  • Point edge 

  • In Euclidean geometry, AAA (Point Edge) (or only AA, since in Euclidean geometry the edges of a triangle mean 180°) does not give data in regards to the span of the two triangles and subsequently demonstrates just closeness and not consistency in Euclidean space. 

  • Be that as it may, in round geometry and hyperbolic geometry (where the aggregate of the points of a triangle fluctuates with size) AAA is adequate for harmoniousness on a given bend of surface.In an Euclidean framework, compatibility is principal; it is the partner of equity for numbers. In systematic geometry, consistency might be characterized naturally in this manner: two mappings of figures onto one Cartesian facilitate framework are consistent if and if, for any two focuses in the primary mapping, the Euclidean separation between them is equivalent to the Euclidean separation between the comparing focuses in the second mapping. 

  • A more formal definition expresses that two subsets An and B of Euclidean space Rn are called consistent if there exists an isometry f : Rn → Rn (a component of the Euclidean gathering E(n)) with f(A) = B. Compatibility is a proportionality connection. 

  • Harmonious conic segments 

  • Two conic segments are compatible if their unconventionalities and one other unmistakable parameter portraying them are equivalent. Their erraticisms build up their shapes, correspondence of which is adequate to set up comparability, and the second parameter then sets up size. Since two circles, parabolas, or rectangular hyperbolas dependably have similar flightiness (particularly 0 on account of circles, 1 on account of parabolas, and {\displaystyle {\sqrt {2}}} {\sqrt {2}} on account of rectangular hyperbolas), two circles, parabolas, or rectangular hyperbolas need one and only other normal parameter esteem, building up their size, for them to be consistent. 

  • Harmonious polyhedra 

  • For two polyhedra with similar number E of edges, similar number of appearances, and similar number of sides on comparing faces, there exists an arrangement of at most E estimations that can set up regardless of whether the polyhedra are congruent. For shapes, which have 12 edges, just 9 estimations are vital. 

  • Consistent triangles on a circle 

  • Fundamental articles: Illuminating triangles § Tackling circular triangles, and Round trigonometry § Arrangement of triangles 

  • Likewise with plane triangles, on a circle two triangles having similar grouping of point side-edge (ASA) are essentially consistent (that is, they have three indistinguishable sides and three indistinguishable angles).This can be viewed as tails: One can arrange one of the vertices with a given edge at the south shaft and run the agree with surrendered length the prime meridian. Knowing both edges at either end of the section of settled length guarantees that the other two sides exude with an extraordinarily decided direction.

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