Question

Which scenario is a counterexample that disproves the AAA criteria for congruence?

A. Two 45°−45°−90° triangles that share a diagonal to form a square.
B. Two right triangles with hypotenuses that are 6 in and 8 in.
C. Two isosceles triangles with bases of 2 ft and 7 ft.
D. Two equilateral triangles with side lengths of 4 cm and 10 cm.

Answer 1

Option D, two equilateral triangles with side lengths of 4 cm and 10 cm, is a counterexample disproving the AAA criterion for congruence because they have equal angles but different side lengths.

Step-by-step explanation:

The AAA (Angle-Angle-Angle) criterion states that two triangles are similar if they have corresponding angles that are equal. However, similarity does not guarantee congruence, which requires equal side lengths as well.

A counterexample that disproves the AAA criterion for congruence would involve two triangles with all equal angles but different side lengths.

The correct counterexample is D. Two equilateral triangles with side lengths of 4 cm and 10 cm. Despite having all angles equal (since all equilateral triangles have angles of 60 degrees), these two triangles are not congruent because their side lengths are different.

The differing side lengths mean that they cannot be overlaid exactly, one on top of the other, which is the definition of congruence.