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Geometry: Congruent Triangles



 



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One way to describe congruent triangles is that two triangles are congruent if one fits on top of the other exactly without any change. The triangles only match if their corresponding parts match such as the sides and angles. In fact, it is necessary that there is a one-to-one correspondence in equality or congruence between each part of one triangle to each part of the other for the two triangles to be congruent.

The symbol used for representing a triangle is triangle.. The two direction arrow, ↔, is used to signify correspondence.

The statement, triangle.ABC ↔ triangle.DEF means that the triangle with vertices ABC corresponds in all its parts with triangle DEF.

There are six pairs of corresponding parts with three pairs being corresponding sides and three pairs corresponding angles.

One definition for congruent triangles can be formally defined as follows.

Congruent Triangles
Two triangles are congruent if there is a one-to-one correspondence between their vertices and each correspondence is congruent.


Given two triangles triangle.ABC and triangle.DEF, the correspondence of parts can be described as:

mangle.A = mangle.D       AB = DE
mangle.B = mangle.E       BC = EF
mangle.C = mangle.F       AC = DF
Corresponding parts of congruent triangles are congruent.

If all these equalities hold for the two triangles in question then the two triangles are congruent. The converse is also true if two triangles are congruent then the measures of their corresponding parts are equal.

Direct Congruence

When trying to determine if two triangles are congruent or not it is also possible to classify them according to two types of congruence: direct or inverse.

With direct congruence, the congruent triangles have the same orientation which means that if one is rotated and moved over and on top of the other one then they would fit exactly without flipping.

Triangles with direct congruence are identical regardless of orientation.

Inverse Congruence

If they have inverse congruence then one would have to be flipped over as well as possibly having to move and rotate it.

Triangles with inverse congruence are reflected versions of each other.

Since inversely congruent triangles can still have corresponding parts, they satisfy the definition for congruent triangles.

Congruent Triangles
Two triangles are congruent when the six parts of the first are congruent with the six parts that correspond to the second.

Another common way of describing congruent triangles is with the statement:

Corresponding parts of congruent triangles are congruent.

This statement is used often in geometry proofs involving congruent triangles and it is often abbreviated as: CPCTC.

Congruence Postulates

It is often possible to determine if two triangles are congruent by determining if three of the corresponding parts are congruent instead of all six.

There are three basic postulates that relate three corresponding parts of one triangle to another:

SSS – Side Side Side
If the three sides of one triangle are congruent to the corresponding sides of another then the triangles are congruent.


A triangle that has all sides congruent to another triangle makes both triangles congruent to each other.
SAS – Side Angle Side
If two sides of a triangle along with the included angle are congruent to the corresponding two sides with included angle in a second then the two triangles are congruent.


If two triangles have two congruent sides and the included angles are congruent, then the triangles are congruent.
ASA – Angle Side Angle
If two angles of a triangle along with the included side are congruent to the two angles with its included side of the other then the two triangles are congruent.


An angle or side is said to be included if it is between two of the same type of part. A side is included by the angles at its endpoints and an angle is included by the sides of the triangle that also form the sides of the angle.

If two angles and the included side of a triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.

It is also possible to determine congruency if the two angles are given along with a side that is not included. This is sometimes stated as the following theorem:

AAS – Angle Angle Side
If two angles and a non-included side of one triangle are congruent to the corresponding angles and the side of the other then the triangles are congruent.


This can also be done by using the sum of angle measures of the triangle which is 180°. When the third angle is found, the ASA postulate can be used.

If the two angles of a triangle are congruent to another triangle, the third angle is also congruent and can be used with an included side to determine the congruency of the triangles.

There is another three part combination of the triangle called:

SSA – Side Side Angle

This involves two sides of a triangle with a non-included angle. With this type of combination, however, it is not possible to determine if two triangles are congruent. This is because the included angle between the two sides can have different measures.

If two triangles have two congruent sides and a congruent non-included angle, it is possible for the triangles to be incongruent.
 




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Congruence
Congruent Polygons





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