Congruence of Triangles

 EXPLANATION ;

Two triangles are said to be congruent if all three corresponding sides are equal and all three corresponding angles are equal in measure.

Symbol - (≡)

Note the difference of the symbol of equality (=) and the congruency (≡).

Explanation; 

Sides : AC=ED

             CB=DE

             AB=AF

Angles : BA^C = DE^F

               EF^D= AB^C

               BC^A= ED^F

1.) First case 

The case in which two sides and the included angle of one Triangle are equal to two sides and the included angle of another triangle .

Example ;

• Showing that two triangles are congruent in the above manner is mentioned in short as being congruent under the case SAS - (SIDE ANGLE SIDE )

 > In the triangles of ABC ∆ and PQR ∆,

The angle AB^C which 30° in called the included angle of the sides AB and CB , Similarly PQ^R is the included angle of the sides PQ and RQ of the triangle PQR .

Proof ; AB = RQ ( data )

             BC = PQ ( data )

             AB^C = RQ^P ( data ) 

therefore ∆ABC ≡ ∆PQR (SAS)

2.) Second case

The case in which the magnitudes of two angles and the length of a side of a triangle are equal to the magnitudes of two angles and the length of a corresponding side of another triangle

Note : 

Corporate sides are defined as those which are opposite equal angles of the triangles

If two angles and a side of one Triangle are equal to two angles and corresponding side of another triangle, then the two triangles are congruent.

That two triangles are congruent under these conditions is started concisely as being congruent according to the AAS case .

In the above two triangles ST and MN are a pair of corresponding sides which are also equal . Observe carefully that they are corresponding sides because they are opposite the angles SU^T and ML^N , which are equal to each other.

In the triangles STU and LMN ;

                    STU=MNL (data)

                    TUS=NLM (data)

                     ST=MN (data)

Therefore ∆STU≡∆MNL (AAS)

3.) Third Case

Theorem :

 The Casr of three sides of a triangle being equal to three sifes of another triangle

If the three sides of a triangle are equal to the three sides of another triangle , then the two triangles are congruent.

That two triangles are congruent under these conditions is started concisely as being congruent according to the SSS case

In the triangles XYZ and DEF ;

XY=DF (data)

YZ=EF (data)

ZX=DE (data)

Therefore ∆XYZ≡∆DEF (SSS)

4.)Fourth case

Theorem :

 The case of the hypothenuse and a side of a right - angled triangle being equal to the hypothenuse and a s ide of another right - angled triangle

Explanation : 

If the lengths of the hypothenuse and a side of a right- angled triangle are equal to the lengths of the hypothenuse and a side of another right - angled triangle, then the two triangles are congruent.

That two triangles are congruent under these conditions is started concisely as being congruent according to the RHS (RIGHT - ANGLE - HYPOTENUSE - SIDE) Case .

In the right - angled triangle ABC and PQR 

               AC = PR (data)

               AB = QR (data)

               AB^C=PQ^R (hypotenuse)

Therefore ∆ABC ≡ ∆PQR (RHS)