Question

question 21 of 26
suppose you wish to apply ssa to a triangle, in order to find an angle measure. also suppose the given side lengths of a triangle are equal. which of the following statements is true?

a. there will be infinitely many solutions for the angle.

b. there will be one solution for the angle.

c. there will be two solutions for the angle.

d. there will be zero solutions for the angle.

Explanation:

1. Recall the SSA (Side - Side - Angle) triangle congruence (or solution) condition. When we have two sides and a non - included angle, the number of solutions depends on the relationship between the sides and the height of the triangle.
2. If two sides of a triangle are equal, let's call the equal sides \(a = b\) and the given angle (non - included or included? In SSA, it's non - included, but if \(a = b\), the triangle is isosceles). If we are applying SSA with two equal sides, consider the Law of Sines: \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Since \(a = b\), then \(\sin A=\sin B\). But if we are using SSA to find an angle, and two sides are equal, the triangle is isosceles, and the angle opposite the equal sides will have a unique solution (because in an isosceles triangle, the base angles are equal, and if we are using SSA with two equal sides, the angle we are solving for will be uniquely determined. There is no ambiguity here as in the general SSA case (where we can have 0, 1, or 2 solutions) because the two sides are equal, so the triangle is isosceles and the angle is uniquely determined. For example, if we have two sides of length \(x\) and a non - included angle, but since the sides are equal, the triangle must be isosceles, so the angle opposite the equal sides (or the given angle) will have only one solution.

Answer:

B. There will be one solution for the angle.