A triangle is a polygon having three sides, three angles, and three vertices.
In geometry, we often compare triangles to determine whether they are exactly alike. Two triangles are said to be congruent when they are identical in both shape and size.
In geometry, we often compare triangles to determine whether they are exactly alike. Two triangles are said to be congruent when they are identical in both shape and size.
Congruence of Triangles:
Two triangles are congruent if their corresponding sides are equal in length and their corresponding angles are equal in measure.If one triangle is placed exactly over the other, all the corresponding sides and angles coincide perfectly. This process is called superposition.
The symbol used to denote congruence is \(\cong\).
Corresponding Parts of Congruent Triangles (CPCT):
When two triangles are congruent, each part of one triangle matches exactly with the corresponding part of the other triangle.Consider the triangle \(ABC\) and \(EFD\).
The corresponding parts are:
| Parts | Corresponding pair |
| Vertices | \(A ↔ D\), \(B ↔ E\), \(C ↔ F\) |
| Sides | \(AB ↔ DE\), \(BC ↔ EF\), \(AC ↔ DF\) |
| Angles | \(∠A ↔ ∠D\), \(∠B ↔ ∠E\), \(∠C ↔ ∠F\) |
Important!
The order in which the vertices are written is extremely important.
Conditions for Congruence:
It is not necessary to measure all the sides and angles every time to prove two triangles congruent.
Certain minimum conditions, called congruence criteria, are sufficient.
The commonly used congruence criteria are:
Certain minimum conditions, called congruence criteria, are sufficient.
The commonly used congruence criteria are:
| Criterion | Statement | Applicable to |
| SSS (Side–Side–Side) | If the three sides of one triangle are equal to the corresponding three sides of another triangle, then the two triangles are congruent. | All triangles |
| SAS (Side–Angle–Side) | If two sides and the included angle (the angle between those two sides) of one triangle are equal to the corresponding two sides and included angle of another triangle, then the triangles are congruent. | All triangles |
| ASA (Angle–Side–Angle) | If two angles and the included side of one triangle are equal to the corresponding two angles and included side of another triangle, then the triangles are congruent. | All triangles |
| RHS (Right angle–Hypotenuse–Side) | If the hypotenuse and one corresponding side of a right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle, then the triangles are congruent. | Right-angled triangles only |
Once two triangles are proved congruent, every corresponding side and angle is automatically equal.