Three proofs that the sum of angles of a triangle is 180

Theorem. The sum of the interior angles of any triangle is 180°.

Here are three proofs for the sum of angles of triangles. Proof 1 uses the fact that the alternate interior angles formed by a transversal with two parallel lines are congruent. Proof 2 uses the exterior angle theorem. Proof 3 uses the idea of transformation specifically rotation.

Proof 1

Construct a line through B parallel to AC. Angle DBA is equal to CAB because they are a pair of alternate interior angle. The same reasoning goes with the alternate interior angles EBC and ACB.

alternate interior angles

Proof 2

This is similar to Proof 1 but the justification used is the exterior angle theorem which states that the measure of the exterior angle of a triangle is the sum of the measures of the two remote interior angles. In the diagram, angle A and angle B are the remote interior angles and angle BCD is the exterior angle. Click this link for the proof of the exterior angle theorem.

exterior angle

Proof 3

Rotate ΔABC 180° clockwise about D the midpoint of BC. This rotation produces the image  ΔA’B’C’. Rotate the image again by 180° clockwise but this time about E, the midpoint of A’B’. In the figure you can see that a+b+c forms a straight line and hence measures 180°.

proof using rotation



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