# Alternate Interior Angles

"Alternate interior angles are equal." Check here for an explanation of alternate interior angles. This is all we need to prove that the sum of the angles in any triangle is 180. Given any triangle, ABC. Then draw a line through A parallel to the side BC, as shown. The two green angles (at A & C) are alternate interior angles, and so they are equal. The two purple angles (at A & B) are alternate interior angles, and so they are equal. The straight angle at A is 180 and is the sum of the green, purple and red angles. But the angles in the triangle are these green, purple and red angles. So the sum of the angles in any triangles is 180.
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