Question

which triangle meets the criteria for the ambiguous case?

Explanation:

Step1: Recall ambiguous case rules

The ambiguous case (SSA) occurs when we have two sides and a non-included angle, where the side opposite the given angle is shorter than the adjacent side, and the given angle is acute.

Step2: Analyze first triangle

This triangle has two angles ($58^\circ, 51^\circ$) and one side (7) → this is AAS, not ambiguous.

Step3: Analyze second triangle

We have two sides (4, 6) and a non-included acute angle ($66^\circ$). The side opposite the $66^\circ$ angle is the unknown side; the adjacent side is 4, and $4 < 6$. This fits SSA with the given angle acute and adjacent side shorter than the other known side, so it is the ambiguous case.

Step4: Analyze third triangle

We have two sides (3, 5) and an included angle ($84^\circ$) → this is SAS, not ambiguous.

Answer:

The second triangle (with sides 4, 6 and non-included angle $66^\circ$)