Notes On Congruence of Triangles - CBSE Class 9 Maths

We come across many figures having the same shape and same size. Such figures are known as congruent figures. 

Two triangles are said to be congruent if all the sides and the angles of one triangle are respectively equal to corresponding sides and angles of other triangle.

Corresponding parts of congruent triangles are equal. This is written in short as 'CPCT'. It is necessary to write the correspondence of vertices in correct order for writing of congruence of triangles in symbolic form. 

Consider the triangles ABC and XYZ. If AB = XY, BC = YZ, AC = XZ and ∠A = ∠X, ∠B = ∠Y, ∠C = ∠Z,  then ΔABC ≅ ΔXYZ.     
                                      


Two triangles are said to be congruent by using any of the following four rules. They are Side Angle Side (SAS) Rule, Angle Side Angle (ASA) Rule, Side Side Side (SSS) Rule, Right angle Hypotenuse Side (RHS) Rule.
  
SAS Congruence Rule
If two sides and included angle of one triangle are equal to the corresponding two sides and included angle of other triangle, then the two triangles are congruent. For two triangles to be congruent, equal angles must be included between the pairs of equal sides. So, SAS congruence rule holds but not ASS or SSA rule.      

Consider triangles ABC and XYZ, If AB = XY, BC = YZ and ∠B = ∠Y, then ΔABC ≅ ΔXYZ.   
                                                        


ASA Congruence Rule 
If two angles and included side of one triangle are equal to two angles and the included side of other triangle, then the two triangles are congruent.      

Consider triangles ABC and XYZ. If BC = YZ, ∠B = ∠Y and ∠C = ∠Z, then ΔABC ≅ ΔXYZ. 
                                     
           
If two pairs of angles of a triangle are equal, then the third pair is also equal since the sum of the three angles of a triangle is 180°. So, two triangles are congruent if any two pairs of angles and one pair of corresponding sides are equal. It is called as the AAS Congruence Rule.

SSS Congruence Rule 
If three sides of one triangle are equal to three sides of the other triangle, then the two triangles are congruent.

Consider triangles ABC and XYZ. If AB = XY, BC = YZ and AC = XZ, then ΔABC ≅ ΔXYZ. 

     

RHS Congruence Rule

Two right angled triangles are said to be congruent if the hypotenuse and one side of one triangle are equal to the hypotenuse and corresponding side of the other triangle. Note that, the right angle is not the included angle between the congruent sides. RHS stands for Right angle - Hypotenuse - Side.

Consider right angled triangles ABC and XYZ. If ∠B = ∠Y = 90°, BC = YZ and AC = XZ, then ΔABC ≅ ΔXYZ. 
                                          

Summary

We come across many figures having the same shape and same size. Such figures are known as congruent figures. 

Two triangles are said to be congruent if all the sides and the angles of one triangle are respectively equal to corresponding sides and angles of other triangle.

Corresponding parts of congruent triangles are equal. This is written in short as 'CPCT'. It is necessary to write the correspondence of vertices in correct order for writing of congruence of triangles in symbolic form. 

Consider the triangles ABC and XYZ. If AB = XY, BC = YZ, AC = XZ and ∠A = ∠X, ∠B = ∠Y, ∠C = ∠Z,  then ΔABC ≅ ΔXYZ.     
                                      


Two triangles are said to be congruent by using any of the following four rules. They are Side Angle Side (SAS) Rule, Angle Side Angle (ASA) Rule, Side Side Side (SSS) Rule, Right angle Hypotenuse Side (RHS) Rule.
  
SAS Congruence Rule
If two sides and included angle of one triangle are equal to the corresponding two sides and included angle of other triangle, then the two triangles are congruent. For two triangles to be congruent, equal angles must be included between the pairs of equal sides. So, SAS congruence rule holds but not ASS or SSA rule.      

Consider triangles ABC and XYZ, If AB = XY, BC = YZ and ∠B = ∠Y, then ΔABC ≅ ΔXYZ.   
                                                        


ASA Congruence Rule 
If two angles and included side of one triangle are equal to two angles and the included side of other triangle, then the two triangles are congruent.      

Consider triangles ABC and XYZ. If BC = YZ, ∠B = ∠Y and ∠C = ∠Z, then ΔABC ≅ ΔXYZ. 
                                     
           
If two pairs of angles of a triangle are equal, then the third pair is also equal since the sum of the three angles of a triangle is 180°. So, two triangles are congruent if any two pairs of angles and one pair of corresponding sides are equal. It is called as the AAS Congruence Rule.

SSS Congruence Rule 
If three sides of one triangle are equal to three sides of the other triangle, then the two triangles are congruent.

Consider triangles ABC and XYZ. If AB = XY, BC = YZ and AC = XZ, then ΔABC ≅ ΔXYZ. 

     

RHS Congruence Rule

Two right angled triangles are said to be congruent if the hypotenuse and one side of one triangle are equal to the hypotenuse and corresponding side of the other triangle. Note that, the right angle is not the included angle between the congruent sides. RHS stands for Right angle - Hypotenuse - Side.

Consider right angled triangles ABC and XYZ. If ∠B = ∠Y = 90°, BC = YZ and AC = XZ, then ΔABC ≅ ΔXYZ. 
                                          

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