Proof method 1: straight line cutting angle method
This is a common method to prove the theorem of triangle interior angle sum, and it is also a relatively intuitive method. We will prove this by cutting the angle in a straight line.
Draw an arbitrary straight line to divide the triangle into two parts. Let's take △ABC as an example.
Now we can see that △ABC is divided into two angles by a straight line: ∠A and ∠ C, and the sum of the degrees of these two angles is equal to 180 degrees.
Similarly, we can see that the straight line is also divided into two triangles: △ABC and △ACB. The sum of the internal angles of these two triangles is also equal to 180 degrees.
Since two triangles share an angle (∠A or ∠C), the sum of their internal angles is equal to 180 degrees.
Therefore, the sum of the internal angles of △ABC and △ACB is equal to 180 degrees respectively.
Since the sum of the internal angles of △ABC and △ACB is the same, the sum of their internal angles is equal to 180 degrees.
In this way, we have proved the theorem of triangle interior angle sum: the triangle interior angle sum is equal to 180 degrees.
Proof method 2: parallel tangent angle method
This is another method to prove the theorem of triangle interior angle sum by using the properties of parallel lines.
Suppose we have a triangle △ABC.
Draw a straight line parallel to BC through point A. ..
According to the properties of parallel lines, we can get the relationship between ∠BAC and ∠EDC, which are corresponding angles, so ∠BAC = ∠EDC.
Similarly, according to the properties of parallel lines, we can get the relationship between ∠ABC and ∠EFD, which is also the corresponding angle, so ∠ABC = ∠EFD.
Now, we can see the relationship between delta △ABC and delta △EDF. They are a pair of triangles with equal corresponding angles, so the sum of their internal angles is equal.
So ∠BAC+∠ABC+∠ACB = ∠EDC+∠EFD+∠DEF.
According to the properties of angular sums ∠BAC+∠ABC+∠ACB = 180 degrees (or π radians), because they are the sum of internal angles of △ABC.
Similarly, ∠EDC+∠EFD+∠DEF is equal to 180 degrees, because they are the sum of the internal angles of △EDF.
So we have proved the theorem of triangle interior angle sum: the triangle interior angle sum is equal to 180 degrees.