Given any triangle:
Rotate so that the longest side lies horizontally along the base.
Extend the base, and draw a parallel line going through the uppermost corner (these two lines are shown in light blue).
The two red angles are the same size (using the well-known “parallel lines” angle result alternate angles, sometimes called “Z-angles“).
The two yellow angles are the same size (Z-angles again!).
The three angles at the top of the diagram form a straight line, and so they add to 180°.
And so red + green + yellow = 180°: proven!
Alternatively, a great demonstration that the angles in any triangle add to 180° is to make the triangle out of paper, then tear off the three corners. Place the torn corners side by side and you will see they line up to make a perfect 180° straight line.