We know that the sum of the internal angles of any polygon is 360 degrees. This is because starting from the vertex and walking along the edge of the polygon will form a closed line. Each line segment intersects two sides to form two angles. Because there are n line segments from a vertex (n is the number of sides of a polygon), there are always 2n angles. Therefore, the sum of the internal angles of a polygon is (n-2)× 180 degrees. For a triangle, n=3, so the sum of its internal angles is (3-2)× 180 degrees = 180 degrees.

This conclusion can also be proved by geometry. In a triangle, we can make a line that passes through the vertex and is parallel to one side. This line divides the triangle into two small triangles. Because of the nature of parallel lines, we know that the angles of these two small triangles are equal. Therefore, the three internal angles of a triangle can be expressed by the angles of two small triangles, and the conclusion is drawn that the sum of the internal angles of the triangle is 180 degrees.

We can also prove that the sum of the internal angles of a triangle is 180 degrees by the external angles. Because the outer angle of a triangle is equal to the sum of its two adjacent inner angles, the sum of the three outer angles of the triangle is 180 degrees, while the sum of the inner and outer angles is 360 degrees, so the sum of the inner angles of the triangle is 360 degrees minus the sum of the outer angles, that is, 180 degrees.

The application of triangle interior angle sum;

1. Calculation: The triangle interior angle sum theorem can be used for calculation. For example, in geography, the area and perimeter of a triangle can be calculated by measuring its three angles. This is very important for drawing maps and determining the shape and size of the earth's surface.

2. Architectural design: In architectural design, the shape and structure of the building can be designed by using the triangle internal angle and theorem to meet the specific functional and aesthetic requirements.

3. Solving geometric problems: The theorem of triangle interior angle sum can be used to solve various geometric problems, such as finding the degree of triangle interior angle and judging whether a triangle is a right triangle.

4. Design of construction machinery: In the design of construction machinery, various mechanical parts and structures, such as bridges and building structures, can be designed and analyzed by using the internal angles and theorems of triangles.

5. Artistic creation: In artistic creation, you can use the inner angle and theorem of triangle to create various works of art, such as painting and sculpture.

6. Physics: In physics, we can use the triangle interior angle theorem to analyze and understand various physical phenomena such as light refraction and reflection.

7. Meteorology: In meteorology, the internal angles and theorems of triangles can be used to analyze and predict weather changes, such as wind direction and air pressure.

8. Navigation: In navigation, the position and direction of the ship can be determined by using the internal angle and theorem of the triangle.

9. Military science: In military science, we can analyze and design all kinds of military equipment and strategies, such as tanks and missiles, by using the triangle interior angle and theorem.

10, Psychology: In psychology, we can analyze and understand people's behavior and way of thinking by using the triangle interior angle sum theorem.