How to prove that the sum of angles in any triangle equals 180 degrees?

**Classic Euclidean proof:** Given $\triangle ABC$, draw a line through $A$ parallel to $BC$. Label the angles: Let the angles at $A, B, C$ be $\alpha, \beta, \gamma$. At vertex $A$, the parallel line creates two angles equal to $\beta$ (alternate interior angle with $\angle B$) and $\gamma$ (alternate interior angle with $\angle C$). These three angles at $A$ ($\beta, \alpha, \gamma$) form a straight line, so $\beta + \alpha + \gamma = 180^\circ$. **On a sphere:** The sum of angles in a spherical triangle exceeds $180^\circ$. For example, a triangle formed by the equator and two meridians meeting at $90^\circ$ has angle sum $270^\circ$. The excess $E = \alpha + \beta + \gamma - \pi$ is proportional to the area: $E = \frac{\text{Area}}{R^2}$. This is how we know Earth is curved!