Question

question 13 of 26
which of the following triangle cases may have one, two, or zero solutions?
a. aaa
b. ssa
c. saa
d. asa

Explanation:

• For option A (AAA): AAA (Angle - Angle - Angle) determines the shape of a triangle (similarity) but not the size. There are infinitely many similar triangles with the same angle measures, but when solving for a specific triangle (e.g., with a given side length), it's not about 0, 1, or 2 solutions in the ambiguous sense. Also, in the context of solving a triangle (finding unknown sides/angles), AAA alone doesn't have the ambiguity of 0, 1, or 2 solutions.
• For option B (SSA): The SSA (Side - Side - Angle) case is the ambiguous case in triangle solving. Let's consider a triangle with two sides and a non - included angle. Let's say we have side \(a\), side \(b\), and angle \(A\) (where \(a\) and \(b\) are two sides and \(A\) is the angle opposite to side \(a\) or not, depending on the configuration). Using the Law of Sines \(\frac{a}{\sin A}=\frac{b}{\sin B}\), we can solve for \(\sin B=\frac{b\sin A}{a}\).
• If \(\frac{b\sin A}{a}> 1\), then there is no solution (since the sine of an angle cannot be greater than 1).
• If \(\frac{b\sin A}{a} = 1\), then \(B = 90^{\circ}\) and there is one solution (a right - triangle).
• If \(\frac{b\sin A}{a}<1\), we may have two solutions for angle \(B\) (an acute and an obtuse angle) or one solution (if the obtuse angle would not form a valid triangle with the given sides and angle).
• For option C (SAA): SAA (Side - Angle - Angle) is a case where we can use the Law of Sines to solve the triangle. Since we know two angles, the third angle is determined (\(A + B+C=180^{\circ}\)), and then using the Law of Sines \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\), we can find the other sides. There is always one unique solution for a SAA triangle (given that the sum of angles is \(180^{\circ}\) and the Law of Sines can be applied consistently).
• For option D (ASA): ASA (Angle - Side - Angle) is also a case where we can find the third angle (\(A + B + C=180^{\circ}\)) and then use the Law of Sines to find the other two sides. There is always one unique solution for an ASA triangle.

Answer:

B. SSA