Given two angle measures and the length of the included side in a triangle, how many distinct triangles can be formed?

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer: (B)

Verified step by step guidance

1. Identify the given elements: two angles and the included side of a triangle. This corresponds to the ASA (Angle-Side-Angle) or AAS (Angle-Angle-Side) case in triangle construction.
2. Recall the triangle uniqueness theorem for ASA or AAS: if two angles and the included side are known, exactly one unique triangle can be formed because the third angle is determined by the sum of angles in a triangle being 180°.
3. Calculate the third angle using the formula \(\text{third angle} = 180^\circ - (\text{angle}_1 + \text{angle}_2)\) to confirm the triangle's angle sum property.
4. Use the Law of Sines or basic geometric construction to verify that the given side length fits uniquely between the two known angles, ensuring only one triangle can be formed.
5. Conclude that with two angles and the included side given, there is exactly one distinct triangle possible.