Final answer:
Ken and Betty may have used the Law of Sines or the Law of Cosines to find relationships between sides and angles in triangles formed by four rays originating from a point. The Law of Sines relates the sides of a triangle to the sine of its opposite angles, while the Law of Cosines relates all three sides of a triangle to the cosine of one of its angles.
Step-by-step explanation:
The question is whether Ken and Betty used the Law of Sines or the Law of Cosines to prove that 'if a, then b' while considering four rays A, B, C, and D originating from point O. The process of using these laws is a part of trigonometry, which is grounded in a set of postulates, and the correct application of these rules leads to consistent and reliable results. The Law of Sines is used to find unknown angles and sides in triangles when we have some of the angles and sides known, whereas the Law of Cosines is more generally applicable by relating the lengths of the sides of a triangle with the cosine of one of its angles, often used in triangles that are not right-angled. In the context provided, one possibility could be that Ken and Betty were working with triangles formed by these rays and needed to find relationships between sides and angles. If Ken used the Law of Sines, he would likely have information about some angles and their opposite sides to apply the law: a/sin A = b/sin B, where a and b are the lengths of sides, and A and B are the angles opposite those sides. On the other hand, if Betty used the Law of Cosines, she would have applied the formula: c² = a² + b² - 2ab cos C, where c is the length of one side of the triangle, and C is the angle opposite side c.