Question

identifying the ambiguous case
which triangle meets the criteria for the ambiguous case?

Explanation:

Step1: Recall ambiguous case rules

The ambiguous case (SSA) occurs when we are given 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 and one side (AAS), which is not the ambiguous case.

Step3: Analyze second triangle

This triangle has two sides and a non-included acute angle (SSA). The side opposite the $66^\circ$ angle is unknown; the given sides are 4 (adjacent to the angle) and 6 (longer side). Since the side adjacent to the acute angle is shorter than the other given side, this fits the ambiguous case criteria.

Step4: Analyze third triangle

This triangle has two sides and the included angle (SAS), which is not the ambiguous case.

Answer:

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