All propositions
Proposition I.32
The angle sum
In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles are equal to two right angles.
The interior angles of a triangle sum to 180°. This is the hinge of Euclidean plane geometry — equivalent, in fact, to the parallel postulate. It fails on a sphere and in hyperbolic space.
Step 1 of 4
A triangle, one side produced
Given · Post. 2
Let ABC be a triangle, and produce BC to D. The exterior angle is ACD. We will show ACD equals A plus B, and therefore A + B + C equals two rights.