The sum of the angles of a triangle is 180 degrees.
Draw triangle ABC. Make BC horizontal, for convenience.
Through point A draw a line parallel to the opposite side, namely BC.
Mark a point L on this line to the left of A, and a point R on this line to the right of A.
∠LAB +∠BAC+∠RAL is a straight angle, and so 180 degrees.
∠LAB=∠ABC, and ∠RAC=∠BCA. These pairs of angles are equal since they are both alternate interior angles which a transversal makes with two parallel lines.
∠LAB=∠ABC since LAR is parallel to BC and AB is a transversal of these two parallel lines.
∠LAB +∠BAC+∠RAC is a straight angle and so
∠ABC+∠BAC+∠BCA is a straight angle, equals may be substituted for equals.
