"Alternate interior angles are equal."[br]Check [b][url=https://www.mathsisfun.com/geometry/alternate-interior-angles.html]here[/url][/b] for an explanation of alternate interior angles.[br]This is all we need to prove that the sum of the angles in any triangle is 180[math]^\circ[/math].[br]Given [u]any[/u] triangle, ABC. Then draw a line through A parallel to the side BC, as shown.[br]The two green angles (at A & C) are alternate interior angles, and so they are equal.[br]The two purple angles (at A & B) are alternate interior angles, and so they are equal.[br]The straight angle at A is 180[math]^\circ[/math] and is the sum of the green, purple and red angles.[br]But the angles in the triangle are these green, purple and red angles.[br]So the sum of the angles in any triangles is 180[math]^\circ[/math].[br]