Is the following statement true or false? If three angles of one triangle are respectively equal to three angles of another triangle, then the triangles are congruent.

Step-by-step solution

1. 1

   Separate the two ideas the statement confuses. Congruent means identical in shape and size — one can be laid exactly on the other. Similar means identical in shape only, with all sides scaled by the same factor. Angles alone can only ever control shape.

2. 2

   Recall the actual congruence criteria. Every valid criterion contains at least one side: SSS, SAS, ASA, AAS, and HL for right triangles. There is deliberately no AAA criterion — its absence is the whole point of this question.

3. 3

   Produce a counterexample. The triangle with sides $3,4,5$ and the triangle with sides $6,8,10$ have exactly the same three angles (both are right triangles, and each side of the second is twice the corresponding side of the first). Their areas are $6$ and $24$, so they are certainly not congruent.

4. 4

   Explain why angles cannot pin down size. Fixing the three angles fixes only two independent quantities, since the third angle is forced by $A+B+C=180^\circ$. A triangle needs three independent measurements to be determined up to congruence, and at least one of them must carry a length.

5. 5

   State the correct version. Equal angles do give a genuine theorem — the AA (or AAA) similarity criterion: the triangles are similar, and their sides are proportional. Congruence follows only if one pair of corresponding sides is also known to be equal. The statement as written is therefore false.