Know about the concept of Congruences

Congruences

What is meant by the term Congruence?

In the field of Geometry, two figures are said to be congruent if and only if the given two figures have the same size and shape and also the angles are similar in some cases. This means that either the position of an object can be changed so it coincides with another object very precisely.

Two given points are said to be congruent if and only if the one given figure can be identically transformed into another figure.

In simple words, if we cut out two shapes or figure on two different pieces of paper and align the two cut figures, if the cut figures tend to match, without giving any extra cut-outs or any part of the paper going out of the cut shaped then the two cut figures are said to be congruent.

Congruent Triangles

Two triangles are said to be congruent when all the three angles of both the given triangles are the same and all the three sides of both the given triangles are the same in value and measurements then the two given triangles are said to be congruent.

For Example-

Let us consider that we have two triangles ∆ABC and ∆LMN. We can only say that both the given triangles that is ∆ABC and ∆LMN are congruent if and only if-

Side AB = Side LM

Side BC = Side MN

Side AC = Side LN

Or the measures of 

Angle A (∠A) = Angle L (∠L) 

Angle B (∠B) = Angle M (∠M) 

Angle C (∠C) = Angle N (∠N) 

Are similar in value.

Rules for the Congruence of Triangles

There are four common rules an individual shall follow while performing congruence tests and problem-solving-

The SSS Test or Rule is also called the Side-Side-Side Test of congruent triangles.

The SAS Test or Rule is also called the Side-Angle-Side Test of congruent triangles.

The ASA Test or Rule is also called the Angle-Side-Angle Test of congruent triangles.

The AAS Test or Rule is also called the Angle-Angle-Side Test of congruent triangles.

The Hypotenuse Rule or Test

Congruency Rules with Explanation

ASA Test or the Angle-Side-Angle Test of congruent triangles

Two given triangles are said to be congruent by the ASA Test of Congruent Triangles if two angles and one side of the given two triangles are equal. The correspondence of the angles and sides of the two triangles must match.

SSS Test or the Side-Side-Side Test of congruent triangles

Two given triangles are said to be congruent by the SSS Test of Congruent Triangles if all the three sides of the given two triangles are equal. The correspondence of the angles and sides of the two triangles must match.

RHS Test or Hypotenuse Rule of congruent triangles

Two given triangles are said to be congruent by the RHS rule or Test of congruent triangles if the hypotenuse and an additional another side apart from the hypotenuse from one given triangle is equal or corresponds to the hypotenuse and an additional another side apart from the hypotenuse of the second given triangle are equal.

SAS Test or the Side-Angle-Side Test of congruent triangles

Two given triangles are said to be congruent by the SAS Test of Congruent Triangles if two sides and one angle of the given two triangles are equal. The correspondence of the angles and sides of the two triangles must match.

AAS Test or the Angle-Angle-Side Test of congruent triangles

Two given triangles are said to be congruent by the AAS Test of Congruent Triangles if two angles and one side of the given two triangles are equal. The correspondence of the angles and sides of the two triangles must match.

Summary and Important Key Points regarding Congruency of a Triangle

1. Two given triangles are said to be congruent if and only if the three sides and three angles of the given two triangles are in correspondence.
2. The five tests or attributes that satisfy the rules of congruency are SSS Test, SAS Test, RHS Test, AAS Test and ASA Test.
3. The congruency for two given shaped or generally a triangle is represented by (≅).
4. Two congruent triangles share an equal area and parameters.

Conclusion

Triangle-based numerical problems can be fun to solve and work on. Congruence teaches an individual the basics of matching and identifying if two given triangles are similar or not. In the field of Geometry, two figures are said to be congruent if and only if the given two figures have the same size and shape and also the angles are similar in some cases.

Geometry is also considered to be the basics of Engineering and Architecture. If an individual wishes to enter and pursue a masters degree in any of these two fields the complete knowledge of congruency, shapes and geometry is a must.