Final answer:
A triangle with angles measuring 40, 98, and 42 degrees exists in Euclidean geometry because the sum is 180 degrees, which is a requirement for triangles in that geometric context. These angles could also represent a triangle in spherical geometry if the curvature of the sphere allows for a triangle with angles summing to 180 degrees.
Step-by-step explanation:
Whether a triangle with angle measures of 40, 98, and 42 degrees exists in Euclidean geometry or spherical geometry can be determined by calculating the sum of the angles. In Euclidean geometry, the sum of angles in a triangle must be exactly 180 degrees. Adding the given angles, 40 + 98 + 42, equals 180 degrees, which aligns with the Euclidean property. Therefore, a triangle with these angle measures exists in Euclidean geometry. In spherical geometry, the sum of the angles in a triangle can be more than 180 degrees because of the curvature of the sphere's surface. However, since the sum of the given angles is exactly 180, this set of angles could also represent a triangle on a sphere where the curvature is such that it does not affect the sum of the angles, making it akin to a triangle on a flat plane. Hence, while these measurements are typical of a triangle in Euclidean geometry, they could also exist in a specific case of spherical geometry.