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468:(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. AAS is equivalent to an ASA condition, by the fact that if any two angles are given, so is the third angle, since their sum should be 180°. ASA and AAS are sometimes combined into a single condition,
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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). This can be seen as follows: One can situate one of the vertices with a given angle at the south pole
413:
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
114:
This diagram illustrates the geometric principle of angle-angle-side triangle congruence: given triangle ABC and triangle A'B'C', triangle ABC is congruent with triangle A'B'C' if and only if: angle CAB is congruent with angle C'A'B', and angle ABC is congruent with angle A'B'C', and BC is congruent
512:
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
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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
107:. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. Therefore two distinct plane figures on a piece of paper are congruent if they can be cut out and then matched up completely. Turning the paper over is permitted.
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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 (SSA, or long side-short side-angle), then the two triangles are congruent. The opposite side is sometimes longer when the corresponding angles are
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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 statement is often used as a justification in elementary geometry proofs when a conclusion of the congruence of parts of two triangles is needed after the congruence of the triangles has been established. For example, if two triangles have been shown to be congruent by the
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applies if the objects have the same shape but do not necessarily have the same size. (Most definitions consider congruence to be a form of similarity, although a minority require that the objects have different sizes in order to qualify as similar.)
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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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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).
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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
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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
525:
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
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to them. The last triangle is neither congruent nor similar to any of the others. Congruence permits alteration of some properties, such as location and orientation, but leaves others unchanged, like
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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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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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sides are congruent if and only if they each have numerically identical sequences (even if clockwise for one polygon and counterclockwise for the other) side-angle-side-angle-... for
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The plane-triangle congruence theorem angle-angle-side (AAS) does not hold for spherical triangles. As in plane geometry, side-side-angle (SSA) does not imply congruence.
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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.
482:(hypotenuse-leg): If two right-angled triangles have their hypotenuses equal in length, and a pair of other sides are equal in length, then the triangles are congruent.
441:(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.
429:(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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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
193:. (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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measurements are not enough if the polyhedra are generic among their combinatorial type. But less measurements can work for special cases. For example,
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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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criteria and a statement that corresponding angles are congruent is needed in a proof, then CPCTC may be used as a justification of this statement.
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1001:, congruence may be defined intuitively thus: two mappings of figures onto one Cartesian coordinate system are congruent if and only if, for
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have 12 edges, but 9 measurements are enough to decide if a polyhedron of that combinatorial type is congruent to a given regular cube.
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measurements that can establish whether or not the polyhedra are congruent. The number is tight, meaning that less than
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Second, draw a vector from one of the vertices of one of the figures to the corresponding vertex of the other figure.
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The orange and green quadrilaterals are congruent; the blue is not congruent to them. All three have the same
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Borisov, Alexander; Dickinson, Mark; Hastings, Stuart (March 2010). "A Congruence
Problem for Polyhedra".
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always have the same eccentricity (specifically 0 in the case of circles, 1 in the case of parabolas, and
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Euclidean distance between the corresponding points in the second mapping.
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An example of congruence. The two triangles on the left are congruent, while the third is
449:. In most systems of axioms, the three criteria – SAS, SSS and ASA – are established as
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Relationship between two figures of the same shape and size, or mirroring each other
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character 'approximately equal to' (U+2245). In the UK, the three-bar equal sign
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lengths of the two pairs of corresponding sides. There are a few possible cases:
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If at any time the step cannot be completed, the polygons are not congruent.
1248:. Addison-Wesley. p. 167. Archived from the original on 29 October 2013
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Symbolically, we write the congruency and incongruency of two triangles
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First, match and label the corresponding vertices of the two figures.
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the rotated figure about this matched side until the figures match.
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Congruence of polygons can be established graphically as follows:
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A symbol commonly used for congruence is an equals symbol with a
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the translated figure about the matched vertex until one pair of
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the first figure by this vector so that these two vertices match.
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In more detail, it is a succinct way to say that if triangles
1307:. Mathematics Textbooks Second Edition. G Bell and Sons Ltd.
1243:"Oxford Concise Dictionary of Mathematics, Congruent Figures"
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if, and only if, one can be transformed into the other by an
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with the same combinatorial type (that is, the same number
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Sufficient evidence for congruence between two triangles in
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Corresponding Parts of
Congruent Triangles are Congruent
1335:. Mathematics Textbooks Second Edition. Bookmark Inc.
817:{\displaystyle {\overline {AC}}\cong {\overline {DF}}}
766:{\displaystyle {\overline {BC}}\cong {\overline {EF}}}
715:{\displaystyle {\overline {AB}}\cong {\overline {DE}}}
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Solving triangles § Solving spherical triangles
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Spherical trigonometry § Solution of triangles
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611:{\displaystyle \triangle ABC\cong \triangle DEF,}
1266:: CS1 maint: bot: original URL status unknown (
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119:are used here to show angle and side equalities.
621:with corresponding pairs of angles at vertices
505:thus allowing the SSS postulate to be applied.
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478:(right-angle-hypotenuse-side), also known as
989:Definition of congruence in analytic geometry
267:are equal in length, and their corresponding
157:are congruent if they have the same diameter.
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1373:Jacobs uses a slight variation of the phrase
150:are congruent if they have the same measure.
955:{\displaystyle \angle BCA\cong \angle EFD.}
143:are congruent if they have the same length.
1080:Two conic sections are congruent if their
907:{\displaystyle \angle ABC\cong \angle DEF}
862:{\displaystyle \angle BAC\cong \angle EDF}
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530:and not congruence in Euclidean space.
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48:. The unchanged properties are called
1540:Interactive animations demonstrating
1305:Revision Course in School mathematics
1005:two points in the first mapping, the
263:are congruent if their corresponding
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1479:Bolin, Michael (September 9, 2003).
127:is often used as follows. The word
1481:"Exploration of Spherical Geometry"
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391:{\displaystyle ABC\ncong A'B'C'}
123:In elementary geometry the word
1157:Congruent triangles on a sphere
337:{\displaystyle ABC\cong A'B'C'}
163:two plane figures are congruent
1333:Geometry for Secondary Schools
455:School Mathematics Study Group
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1407:American Mathematical Monthly
1362:, W.H. Freeman, p. 160,
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1284:. Math Open Reference. 2009
1114:{\displaystyle {\sqrt {2}}}
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1575:Equivalence (mathematics)
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1217:Euclidean plane isometry
1076:Congruent conic sections
568:are congruent, that is,
1550:Congruent line segments
1197:, corresponding to the
249:Congruence of triangles
168:The related concept of
1556:at Math Open Reference
1453:"A Congruence Problem"
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405:Determining congruence
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64:if they have the same
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1465:on November 11, 2013.
1384:"Congruent Triangles"
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1068:. Congruence is an
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1527:Cut-the-Knot
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1282:"Congruence"
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553:stands for
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97:translation
95:, namely a
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1564:Categories
1390:2014-02-04
1228:References
1191:above it,
528:similarity
414:triangles.
253:See also:
205:sides and
170:similarity
105:reflection
50:invariants
1421:0811.4197
1131:polyhedra
1091:parabolas
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362:≆
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261:triangles
221:Translate
187:perimeter
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125:congruent
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1488:Archived
1359:Geometry
1331:(2002).
1262:cite web
1222:Isometry
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1129:For two
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234:matches.
209:angles.
103:, and a
101:rotation
89:isometry
58:geometry
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1438:8166476
1199:Unicode
1087:circles
1014:subsets
551:acronym
470:AAcorrS
457:system
239:reflect
226:Third,
155:circles
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1252:2 June
1165:, and
661:; and
637:; and
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269:angles
228:rotate
148:angles
81:points
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1189:tilde
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1139:faces
1093:, or
993:In a
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549:This
545:CPCTC
265:sides
129:equal
66:shape
1364:ISBN
1337:ISBN
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1290:2017
1268:link
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283:△
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