Final answer:
Kayli's observation about the constant ratio in right triangles with a 50° angle is supported by the concept of trigonometric ratios, specifically the sine, which is consistent for a given angle across all similar right-angle triangles.
Step-by-step explanation:
Kayli's conjecture that the ratio of the length of the opposite leg to the hypotenuse in right triangles with a 50° angle is the same is supported by the concept of trigonometric ratios. Specifically, for a fixed angle in a right triangle, the ratios of the sides are consistent, which is the definition of trigonometric functions such as sine, cosine, and tangent.
For example, the sine of an angle θ in a right triangle is the ratio of the length of the side opposite to the angle to the length of the hypotenuse. Thus, if we have a 50° angle, the ratio Kayli noticed, √(15.22/20), could be expressed as sin(50°), which is constant for all right triangles with that angle.
The values from the Pythagorean theorem and trigonometric functions should agree because they are based on fixed ratios that define right-angled triangles. Therefore, regardless of the different sizes of similar right-angled triangles, the sin of a fixed angle is constant, which supports Kayli's conjecture.