Question

which of the following will always determine exactly one triangle? choose all that apply. aaa information asa information saa information ssa information

Explanation:

• AAA (Angle - Angle - Angle): Knowing only the three angles of a triangle does not fix the side lengths. There are infinitely many similar triangles (same - shaped but different - sized) that have the same set of angles. So, AAA information does not determine exactly one triangle.
• ASA (Angle - Side - Angle): If we know two angles and the included side of a triangle, by the Angle - Side - Angle congruence criterion, there is exactly one triangle that can be formed.
• SAA (Side - Angle - Angle): Since the sum of the angles in a triangle is \(180^{\circ}\), if we know two angles (say \(\angle A\) and \(\angle B\)) and a non - included side (say side \(a\)), we can find the third angle \(\angle C=180^{\circ}-(\angle A + \angle B)\). Then, using the Law of Sines \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\), there is exactly one triangle that can be formed.
• SSA (Side - Side - Angle): This is the ambiguous case. Using the Law of Sines \(\frac{a}{\sin A}=\frac{b}{\sin B}\), if we know side \(a\), side \(b\), and angle \(A\) (where \(a\) and \(b\) are the sides, and \(A\) is the non - included angle), there may be no triangle (if \(a\lt b\sin A\)), one triangle (if \(a = b\sin A\) or \(a\geq b\) when \(A\) is acute, or \(a\geq b\) when \(A\) is obtuse), or two triangles (if \(b\sin A\lt a\lt b\) when \(A\) is acute).

Answer:

B. ASA information, C. SAA information