The congruency of two or more triangles depends on the measurements of their angles and sides. The sides and angles of a triangle will determine its type and shape. If two triangles are congruent means their corresponding angles and sides are equal in measure. Students will learn other criteria for congruency of triangles in the following sections with detailed explanations.
What is Triangles Congruence?
Whenever one triangle has three angles, three sides are equal to corresponding angles, corresponding sides of the second triangle, then both triangles are congruent to each other.
Here, △ABC ≅ EFG
Because AC = EG, AB = EF, BC = FG and ∠A = ∠E, ∠F = ∠B, ∠C = ∠G
In the above figure corresponding vertices are A ≅ E, B ≅ F, C ≅ G.
Criteria for Congruency in Triangles
There are some conditions to say that two triangles are congruent. Triangles must and should satisfy those criteria to be congruent. CPCT means Corresponding Parts of Congruent Triangles. The congruency of triangles can be predicted without measuring their dimensions by following these criteria for congruency. They are along the lines:
- SSS (Side Side Side) Criterion
- SAS (Side Angle Side) Criterion
- AAS (Angle Angle Side) Criterion
- ASA (Angle Side Angle) Criterion
- RHS (Right angle Hypotenuse Side) Criterion
- Congruency of Triangles
- Classification of Triangles
- Properties of Angles of a Triangle
- Geometry and Measurement
SSS (Side-Side-Side) Criterion
If all three corresponding sides of triangles are equivalent, then two triangles are congruent by SSS Criterion. SSS Criterion stands for Side-Side-Side Criterion.
△DFG ≅ JHK by SSS Criterion.
Corresponding sides are DG = HK, DF = JK, FG = JK.
SAS (Side-Angle-Side) Criterion
SAS Criterion means Side-Angle-Side Criterion. Two triangles are said to be congruent if two sides and included angle of one triangle is equal to the corresponding sides and included angle of another triangle by SAS rule.
△LMN ≅ NQP by SAS Criterion.
Corresponding two sides are LM = NQ, MN = QP
Corresponding angle ∠M = ∠Q.
AAS (Angle-Angle-Side) Criterion
If the first triangle side is equal to the second triangle side and the corresponding angles of triangles are also equal, then they are congruent by AAS rule. AAS Criterion stands for Angle-Angle-Side Criterion.
△ABC ≅ QRS by AAS Criterion.
Corresponding two angles are ∠B = ∠R, ∠A = ∠Q
The corresponding side is AC = QS
ASA (Angle-Side-Angle) Criterion
Two triangles are congruent if any two angles and the side included between them of one triangle are equal to the corresponding angles and included side of the other triangle by ASA rule. ASA Criterion is Angle-Side-Angle Criterion.
△ABC ≅ DEF by ASA Criterion.
Corresponding angles ∠A = ∠F, ∠C = ∠E
Corresponding side AC = EF
RHS (Right angle-Hypotenuse-Side) Criterion
If the hypotenuse side, any other side of the first right-angled triangle is equal to the hypotenuse side, another corresponding side of the other right-angled triangle, then two triangles are said to be congruent by RHS rule. RHS criterion is the Right angle-Hypotenuse-Side Criterion.
△PQR ≅ XYZ by RHS Criterion.
Corresponding right angle: ∠Q = ∠Y
Corresponding sides: PR = XZ, PQ = XY
Prove the following triangles are congruent and identify the criterion.
At first, identify the corresponding side, angles.
Corresponding side: KM = KM
Corresponding angles: ∠L = ∠N, ∠K = ∠M
Here, Two angles and side of one triangle is equivalent to another triangle.
So, △KLM ≅ LMN by AAS Criterion.
Identify the rule of congruence in the triangles ABC and DCB, and prove that they are congruent triangles.
Let us identify the corresponding sides, angles.
Corresponding sides: PQ = XZ = 3 in and PR = YZ = 5 in
Corresponding angle: ∠Q = ∠X = 90 degrees
Here, two triangles are right angles triangles. The hypotenuse and one side of the first right-angled triangle are equal to the second triangle hypotenuse and side.
△PQR ≅ XYZ by RHS Criterion.
FAQ’s on How to Prove Triangles Congruent
1. What is the full form of CPCT?
CPCT stands for Corresponding parts of Congruent Triangles. It states that if two or more triangles that are congruent to each other are taken then the corresponding angles, sides of the triangles are also congruent.
2. Is AAA criteria for congruence?
AAA (angle-angle-angle) does not work because it can produce similar angles but not congruent triangles.
3. What is the SSS rule?
If all three sides of one triangle are equivalent to the corresponding three sides of the second triangle, then those triangles are congruent by the SSS rule.
4 What are the 4 rules of congruency?
The 4 different congruency of triangles rules is SAS (Side-Angle-Side) rule, SSS (Side-Side-Side) rule, AAS (Angle-Angle-Side) rule, and RHS (Right angles Hypotenuse Side) rule.