Spherical triangles – identify the incorrect statement In a spherical triangle on the surface of a sphere, which one of the following statements is incorrect?

Introduction / Context:
Spherical trigonometry governs angles and sides of triangles drawn on a sphere. Unlike plane triangles, spherical triangles have curved sides (great circle arcs), and their geometric properties differ in key ways, which surveyors and astronomers must remember for accurate computations on the celestial sphere and in geodetic work.

Given Data / Assumptions:

• Triangle lies on a sphere and sides are measured as central angles (radians or degrees).
• Angles are dihedral angles between great circles at the triangle’s vertices.
• Right angle = 90°; π radians = 180°.

Concept / Approach:
In a spherical triangle, each interior angle is strictly between 0° and 180°. The sum of the three angles exceeds 180° (two right angles) by an amount called the spherical excess and is strictly less than 540° (six right angles). There are also special relational properties connecting sums of sides with sums of opposite angles.

Step-by-Step Solution:
Check option (a): Each spherical angle < 180° → true.Check option (b): Sum of angles = 180° → false for spherical triangles; true only for plane triangles.Check option (c): Angle sum lies between 180° and 540° → true.Check option (d): If a + b = π, then A + B = π → a known spherical identity → true.Check option (e): Triangle inequality on a sphere (each side < sum of other two) → true.

Verification / Alternative check:
The spherical excess E satisfies A + B + C = π + E (in radians) with E > 0 for any non-degenerate spherical triangle. Hence equality to 180° cannot hold.

Why Other Options Are Wrong:

• (a), (c), (d), (e) are valid properties of spherical triangles.
• (b) is the planar-triangle rule and is therefore incorrect on a sphere.

Common Pitfalls:
Transferring plane-triangle rules directly to spherical geometry; forgetting that spherical excess makes the angle sum greater than 180°.

Final Answer:
The sum of the three angles is equal to two right angles (exactly 180°)