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See diagram.

Draw ABC as any triangle.

Now, draw a straight line parallel to BC through A.  
     Now angle DAG = angle BAC - opposite angles at an intersection of two lines.

     Angle DAE = angle ABC  as they are angles formed by a line AB intersecting 
       two parallel lines  - included angle.
     Angle GAF = angle EAC - opposite angles at intersection of two lines at A
     Angle GAF = angle BAC    as BC and FE are parallel. AC is intersecting both 
             lines. These are included angles.

     Sum of these angles at A =  angles DAG + GAF + EAD = 180 degrees.
      At any point the total angle 360° degrees is divided into 180° and 180° by a 
      straight line.

So sum of angles A+B+C = 180° degrees

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