CLASS 7TH | UNIT 4: GEOMETRY | CONGRUENCY: CONGRUENT TRIANGLES | REVISION NOTES

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 ≅≅ 

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 P1P1 is congruent to plane P2P2 then it is written or denoted as P1≅P2P1≅P2 

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 AB¯¯¯¯¯¯¯¯AB¯ is congruent to plane CD¯¯¯¯¯¯¯¯CD¯ then it is written or denoted as AB¯¯¯¯¯¯¯¯≅CD¯¯¯¯¯¯¯¯AB¯≅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 ∠ABC∠ABC is congruent to angle ∠CDE∠CDE then it is written or denoted as ∠ABC≅∠CDE∠ABC≅∠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 ΔABCΔABC and ΔPQRΔPQR has correspondence as 

A↔PA↔P

B↔QB↔Q

C↔RC↔R

It can be written as ABC↔PQRABC↔PQR which implies that ΔABC≅ΔPQRΔABC≅ΔPQR 

• In the above case if we denote it as ΔABC≅ΔPRQΔABC≅ΔPRQ then it is not possible since vertex BB is corresponding to vertex QQ not RR 
• 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:

(Image will be uploaded soon)

In triangles ΔABCΔABC and ΔPQRΔPQR

AB=PQAB=PQ

BC=QRBC=QR

AC=PRAC=PR

Then by SSS congruency ΔABC≅ΔPQRΔABC≅ΔPQR

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

(Image will be uploaded soon)

In triangles ΔABCΔABC and ΔPQRΔPQR

AB=PQAB=PQ

∠BAC=∠QPR∠BAC=∠QPR

AC=PRAC=PR

Then by SAS congruency ΔABC≅ΔPQRΔABC≅ΔPQR

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

(Image will be uploaded soon)

In triangles ΔABCΔABC and ΔPQRΔPQR

AB=PQAB=PQ

∠BAC=∠QPR∠BAC=∠QPR

∠ABC=∠PQR∠ABC=∠PQR

Then by ASA congruency ΔABC≅ΔPQRΔABC≅ΔPQR

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

(Image will be uploaded soon)

In triangles ΔABCΔABC and ΔPQRΔPQR

AC=PRAC=PR

BC=QRBC=QR

Then by RHS congruency ΔABC≅ΔPQR