Which option is NOT a valid congruence criterion for triangles?

Which option is NOT a valid congruence criterion for triangles?

• A. AAA
• B. SSS
• C. SAS
• D. ASA

Answer: (A)

Two triangles are congruent when they match exactly in size and shape. That requires information that fixes both how big they are and how their sides and angles relate. The standard congruence criteria—three sides, two sides with the included angle, or two angles with the included side—each provides enough information to lock in both dimensions, so the triangles are identical in every aspect. Using only angle measures, as AAA does, tells you the triangles have the same angles, so they are similar, not necessarily congruent. You can scale one triangle up or down and still preserve the same angles, which changes the side lengths while keeping the angles the same. A concrete example is a 3-4-5 right triangle and a 6-8-10 right triangle: they have the same angles but different sizes, so they aren’t congruent. So the option that does not determine congruence is the one based on equal angles alone.

Two triangles are congruent when they match exactly in size and shape. That requires information that fixes both how big they are and how their sides and angles relate. The standard congruence criteria—three sides, two sides with the included angle, or two angles with the included side—each provides enough information to lock in both dimensions, so the triangles are identical in every aspect.

Using only angle measures, as AAA does, tells you the triangles have the same angles, so they are similar, not necessarily congruent. You can scale one triangle up or down and still preserve the same angles, which changes the side lengths while keeping the angles the same. A concrete example is a 3-4-5 right triangle and a 6-8-10 right triangle: they have the same angles but different sizes, so they aren’t congruent.

So the option that does not determine congruence is the one based on equal angles alone.