If two sides and the angle between them of one triangle are equal to the corresponding two sides and the angle between them of another triangle, then the triangles are congruent. it is known as SAS axiom.
Figure a
Figure b
Bases
In triangle ABC
In triangle PQR
Result
Conditions of construction
AB =
PQ =
∠ABC =
∠PQR =
BC =
QR =
Parts of examination
AC =
PR =
∠BAC =
∠QPR =
∠ACB =
∠PRQ =
Conclusion : Here, two sides and included angle of triangle ABC are equal to the two sides and included angle of triangle PQR. The corresponding sides and angles are also equal. So triangle ABC and triangle PQR are CONGRUENT TRIANGLES.
📐Right angle hypotenuse side (RHS).
If the hypotenuse and one side of a right angled triangle are equal to the hypotenuse and one side of another right angled triangle, then the triangles are congruent by RHS.
Figure a
Figure b
Bases
In triangle ABC
In triangle PQR
Result
Conditions of construction
BC =
QR =
∠ABC =
∠PQR =
AC =
PR =
Parts of examination
AB =
PQ =
∠BAC =
∠QPR =
∠ACB =
∠PRQ =
Conclusion : Here, In right angles triangle ABC and triangle PQR, right angled, hypotenuse and one side of another right angled triangle, then the triangles are CONGRUENT TRIANGLES.
📐Angle Side and Angle (ASA).
If two angles and a side between them of one triangle are equal to the two angles and a side between them of another triangle, then the triangles are congruent. it is known as ASA axiom.
Figure a
Figure b
Bases
In triangle ABC
In triangle PQR
Result
Conditions of construction
∠ABC =
∠PQR =
BC =
QR =
∠ACB =
∠PRQ =
Parts of examination
AC =
PR =
AB =
PQ =
∠CAB =
∠RPQ =
Conclusion : Here, two angles and the included side of one triangle ABC are equal to the two angles and included side of triangle PQR. The corresponding sides and angles are also equal. So triangle ABC and triangle PQR are CONGRUENT TRIANGLES.
📐Side Side Side (SSS).
If the three sides of one triangle are individually equal to the three sides of another triangle, then we can say that the triangles are congruent by SSS.
Figure a
Figure b
Bases
In triangle ABC
In triangle PQR
Result
Conditions of construction
AB =
PQ =
BC =
QR =
AC =
PR =
Parts of examination
∠BAC =
∠QPR =
∠CBA =
∠RQP =
∠ACB =
∠PRQ =
Conclusion : Here, all sides of triangle ABC are separately equal to all sides of triangle PQR. The corresponding angles are also equal. So triangle ABC and triangle PQR are CONGRUENT TRIANGLES.
📐Angle Angle and Side (AAS).
If a side of one triangle along with an adjacent angle, is equal to a side of another triangle, along with its adjacent angle, then the triangles are considered congruent when their corresponding parts are equal in order, i.e., when the remaining corresponding angle and the side opposite to it are also equal.
Figure a
Figure b
Bases
In triangle ABC
In triangle EFG
Result
Conditions of construction
∠ABC =
∠EFG =
∠ACB =
∠FGE =
AC =
EG =
Parts of examination
BC =
FG =
AB =
EF =
∠CAB =
∠GEF =
Conclusion : Here, two angles and a non-included side of triangle ABC are equal to the two angles and a non-included side of triangle EFG. The corresponding sides and angles are also equal. So triangle ABC and triangle EFG are CONGRUENT TRIANGLES.