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
- 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
- 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
- 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
- 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