Question
List all five congruence criteria for triangles. How do you decide which criterion to use in a given problem? Why is SSA (or ASS) not a valid congruence criterion?
(CBSE Classes 7-9 — congruence proofs carry 5-8 marks in boards)
Solution — Step by Step
If all three sides of one triangle are equal to the corresponding three sides of another triangle, the triangles are congruent.
When to use: When the problem gives you all three side lengths of both triangles, or you can prove all three sides are equal using other properties.
If two sides and the included angle (the angle BETWEEN the two sides) of one triangle are equal to the corresponding parts of another triangle, they are congruent.
When to use: When you know two sides and the angle between them. This is the most commonly used criterion in CBSE proofs.
ASA: Two angles and the included side (the side between the two angles) are equal.
AAS: Two angles and a non-included side are equal.
Both are valid because if two angles are known, the third is automatically determined (angle sum = 180°). So ASA and AAS are essentially equivalent.
Applicable ONLY to right triangles. If the hypotenuse and one other side of a right triangle are equal to the corresponding parts of another right triangle, they are congruent.
When to use: When the problem involves right triangles and mentions the hypotenuse.
flowchart TD
A["Two triangles: are they congruent?"] --> B{What information is given?}
B -->|All 3 sides| C["SSS"]
B -->|2 sides + included angle| D["SAS"]
B -->|2 angles + included side| E["ASA"]
B -->|2 angles + any side| F["AAS"]
B -->|Right triangle + hypotenuse + side| G["RHS"]
B -->|2 sides + non-included angle| H["NOT valid!<br/>(SSA ambiguous case)"]
style H fill:#ff6b6b,stroke:#333Why This Works
Congruence criteria are the minimum information needed to uniquely determine a triangle. Three independent measurements (properly chosen) fix the shape and size completely. SSS, SAS, ASA, AAS, and RHS each provide three constraints that leave no ambiguity.
SSA (two sides and a non-included angle) fails because it can produce two different triangles — the “ambiguous case.” The side opposite the given angle can swing to two positions, creating two valid triangles. That is why SSA is not a congruence criterion.
Alternative Method
For CBSE proofs, follow this approach: (1) Mark the given equal parts on the diagram. (2) Look for additional equal parts from the figure (common sides, vertically opposite angles, alternate angles). (3) Choose the criterion that matches the three parts you have established. Almost always, you need to prove three things equal to use a criterion.
Common Mistake
Students use AAA (three angles equal) as a congruence criterion. AAA proves similarity, NOT congruence. Two triangles with all three angles equal have the same shape but can be different sizes. You need at least one side to establish congruence. Similarly, SSA (two sides and a non-included angle) is NOT valid — it can give two different triangles. Only SSS, SAS, ASA, AAS, and RHS are accepted.