To determine which of the statements is incorrect in relation to a spherical triangle, let's analyze each one:

1. 180° < A + B + C < 540°

• This is correct. For a spherical triangle, the sum of the angles $A + B + C$ is greater than $180^\circ$ but less than $540^\circ$.

2. A + B < C + 180°, A + C < B + 180°, B + C < A + 180°

• This is also correct. These inequalities reflect the relationships among the angles in a spherical triangle.

3. a + b > c; a + c > b; b + c > a

• This is correct. These are the triangle inequalities for the sides $a, b, c$ of a spherical triangle.

4. 0° < a + b + c < 360°

• This statement is incorrect. The sum of the sides of a spherical triangle (measured as angles) must satisfy $a + b + c > 180^\circ$. However, it is incorrect to state $a + b + c < 360^\circ$ without context, as the total can exceed $360^\circ$ depending on the triangle.

Correct Answer:

4. $0° < a + b + c < 360°$

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Expanding Questions:

1. What are the key properties of a spherical triangle compared to a Euclidean triangle?
2. How does the sum of angles in a spherical triangle relate to the curvature of the sphere?
3. Can a spherical triangle have a side greater than 180°? Why or why not?
4. How are the sides of a spherical triangle measured differently from a plane triangle?
5. What are the implications of angle excess in spherical geometry?

Tip:

In spherical geometry, the angle excess ($A + B + C - 180^\circ$) directly