Questions: Which scenario is a counterexample that disproves the AAA criteria for congruence? A. Two equilateral triangles with side lengths of 4 cm B. Two isosceles triangles with congruent non base angles and bases of 2 ft and 7 ft C. Two 45°-45°-90° triangles that share a diagonal to form a square D. Two right triangles with hypotenuses that are 6 in and 8 in E. I don't know yet

Solution Steps

To find a counterexample that disproves the AAA (Angle-Angle-Angle) criteria for congruence, we need to identify a scenario where two triangles have the same angles but are not congruent. Congruent triangles must have the same size and shape, meaning all corresponding sides are equal. The AAA criteria only ensure similarity, not congruence, as it does not account for the size of the triangles.

The AAA (Angle-Angle-Angle) criteria for triangle congruence states that if two triangles have equal corresponding angles, they are similar but not necessarily congruent. This means that two triangles can have the same shape but different sizes.

We need to evaluate the given scenarios to find a counterexample that demonstrates two triangles with the same angles but different side lengths, thus disproving the AAA criteria for congruence.

• Scenario A: Two equilateral triangles with side lengths of \(4 \, \text{cm}\) are congruent since all sides and angles are equal.
• Scenario B: Two isosceles triangles with congruent non-base angles and bases of \(2 \, \text{ft}\) and \(7 \, \text{ft}\) have the same angles but different bases, making them not congruent.
• Scenario C: Two \(45^{\circ}-45^{\circ}-90^{\circ}\) triangles that share a diagonal form a square, which are congruent.
• Scenario D: Two right triangles with hypotenuses of \(6 \, \text{in}\) and \(8 \, \text{in}\) are not congruent as they have different hypotenuse lengths.

From the analysis, Scenario B is the only case where two triangles have the same angles but different side lengths, thus serving as a counterexample to the AAA criteria for congruence.

Final Answer