Congruence of Triangles Class 7 Notes CBSE Maths Chapter 7 [Free PDF Download]

Congruence

• Two object is said to be congruent if and only if their shape and size are same.

• The relation between two congruent object is called congruence. 

• Congruence is denoted by symbol $\cong $ 

Congruence of Plane Figures

• Two plane figures are said to be congruent when two plane figures superpose each other i.e., when one figure is placed on another figure they coincide. 

• Suppose a plane ${{P}_{1}}$ is congruent to plane ${{P}_{2}}$ then it is written or denoted as ${{P}_{1}}\cong {{P}_{2}}$ 

Congruence Among Line Segments

• Two lines are said to be congruent if they have equal length.

• Conversely, if two lines are congruent, they are of equal length.

• Suppose a line $\overline{AB}$ is congruent to plane $\overline{CD}$ then it is written or denoted as $\overline{AB}\cong \overline{CD}$

Congruence of Angle

• Two angles are said to be congruent if they have equal measure. 

• Conversely, if two angles are congruent, they are of equal measure. 

• Suppose an angle $\angle ABC$ is congruent to angle $\angle CDE$ then it is written or denoted as $\angle ABC\cong \angle CDE$

Congruence of Triangle

• Two triangles are said to be congruent if they have corresponding vertices, corresponding sides and corresponding angles equal.

• Let us suppose two triangles $\Delta ABC$ and $\Delta PQR$ has correspondence as 

$ A\leftrightarrow P $

$ B\leftrightarrow Q $

$ C\leftrightarrow R $

It can be written as $ABC\leftrightarrow PQR$ which implies that $\Delta ABC\cong \Delta PQR$ 

• In the above case if we denote it as $\Delta ABC\cong \Delta PRQ$ then it is not possible since vertex $B$ is corresponding to vertex $Q$ not $R$ 

• Therefore, congruence should be denoted in proper way since the order of the letters in the names of congruent triangles displays the corresponding relationships.

Criteria For Congruence of Triangle 

• There are four criteria for congruence of triangle and these are

1. SSS Congruence Criteria

• If under a given correspondence, the three sides of one triangle are equal to the three corresponding sides of another triangle, then the triangles are congruent.

• For example:

In triangles $\Delta ABC$ and $\Delta PQR$

$ AB=PQ $

$ BC=QR $

$ AC=PR $

Then by SSS congruency $\Delta ABC\cong \Delta PQR$

2. SAS Congruence Criteria

• If under a correspondence, two sides and the angle included between them of a triangle are equal to two corresponding sides and the angle included between them of another triangle, then the triangles are congruent.

• For example

In triangles $\Delta ABC$ and $\Delta PQR$

$ AB=PQ $

$ \angle BAC=\angle QPR $

$ AC=PR $

Then by SAS congruency $\Delta ABC\cong \Delta PQR$

3. ASA Congruence Criteria

• If under a correspondence, two angles and the included side of a triangle are equal to two corresponding angles and the included side of another triangle, then the triangles are congruent.

• For example

In triangles $\Delta ABC$ and $\Delta PQR$

$ AB=PQ $

$ \angle BAC=\angle QPR $

$ \angle ABC=\angle PQR $

Then by ASA congruency $\Delta ABC\cong \Delta PQR$

4. RHS Congruence Criteria

• If under a correspondence, the hypotenuse and one side of a right-angled triangle are respectively equal to the hypotenuse and one side of another right-angled triangle, then the triangles are congruent.

• This criterion is applicable on right angle triangles only.

• For example

In triangles $\Delta ABC$ and $\Delta PQR$

$ AC=PR $

$ BC=QR $

Then by RHS congruency $\Delta ABC\cong \Delta PQR$