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In order to prove that the sum of the internal angles of a triangle is equal to 180, that is, the sum of the three internal angles of a triangle is a right angle, an auxiliary line needs to be made to convert the sum of the three internal angles of a triangle into the sum of the three angles that make up a right angle.

Here are some circular arguments:

1. Fold the three angles of a triangle inward, and the three angles just form a right angle, so it is 180 degrees.

2. Make a parallel line at a vertex on the other side and prove it with an inner angle.

3.

Make a triangle ABC

Draw a straight line through point A, and EF is parallel to BC.

Angle EAB= angle b

Angle FAC= angle c

Angle EAB+ Angle FAC+ Angle BAC= 180.

Angle BAC+ Angle B+ Angle C= 180.

4. Interior Angle and Formula (n-2)* 180

5. Let three vertices of a triangle be A, B and C, corresponding to Angle A, Angle B and Angle C respectively; The straight line L passing through point A is parallel to the straight line BC, and the angle formed by L and ray AB is b', the angle formed by L and ray AC is c', and the angles b' and b, and c' and c respectively form internal angles. According to the equality theorem of internal angles of parallel lines, it can be concluded that the sum of internal angles of triangle = angle A+ angle B+ angle C= angle A+ angle B'+ angle C' = 65433.

6. extend the side ABC of the triangle, DAB = C+B, EBA = A+C, FCA = a+b.

So DAB+EBA+FCA=2A+2B+2C=360 (the sum of the outer angles of the triangle is 360).

So A+B+C= 180.

7. Extend one side of the triangle to form triangular diplomacy. It is easy to find that this angle and the inner angle of the adjacent triangle add up to a flat angle (180 degrees), so they are adjacent complementary angles. Then draw a straight line parallel to the opposite side of this angle from the vertex of this inner angle and divide that diplomacy into two angles. It can be proved that the other two angles of the triangle are equal to the two angles of this diplomatic division by using two straight lines parallel, at the same angle and at the same angle. Then the sum of three internal angles of a triangle is equal to that internal angle plus its adjacent complementary angle, which is 180 degrees.

8. Mark three triangles with the same size with the letters A, B and C at the positions corresponding to the three angles, and then put the A angle of the first triangle, the B angle of the second triangle and the C angle of the third triangle together so that their lower sides (or upper sides) just form a straight line, that is, the three angles form a right angle. That is to say, the sum of the degrees of the three angles is 180 degrees.