In any triangle, the three interior angles always add up to exactly 180 degrees. This is one of the most fundamental properties in geometry. Let's consider a triangle with vertices A, B, and C, and examine why the sum of angles A, B, and C equals 180 degrees.
Now we begin the proof. Step one: draw a line through vertex A that is parallel to side BC. Let's call this line DE, with point D on the left side of A and point E on the right side. This parallel line is the key to our proof, as it will help us establish important angle relationships.
Step two: identify the alternate interior angles. When line AB acts as a transversal crossing the parallel lines DE and BC, it creates alternate interior angles. Angle DAB equals angle ABC. Similarly, when line AC crosses the parallel lines, angle EAC equals angle ACB. These equal angle pairs are crucial for our proof.
Step three: use the property that angles on a straight line sum to 180 degrees. The three angles DAB, BAC, and EAC all lie on the straight line DE at point A. Since angles on a straight line always sum to 180 degrees, we have: angle DAB plus angle BAC plus angle EAC equals 180 degrees.