Welcome to our lesson on polygon interior angles. A polygon is a closed figure made up of straight line segments. We can see examples like triangles with 3 sides and quadrilaterals with 4 sides. Each polygon has interior angles formed where two sides meet at a vertex.
Let's examine a triangle more closely. Every triangle has three interior angles. We can label these angles as alpha, beta, and gamma. No matter what shape the triangle takes, whether it's acute, right, or obtuse, the sum of these three interior angles is always exactly 180 degrees.
Here's a visual proof of why triangle angles sum to 180 degrees. We draw a line parallel to the base of the triangle through the top vertex. This creates a straight line, which we know has angles that sum to 180 degrees. Due to the properties of parallel lines, the angles formed are equal to the original triangle's angles, proving our formula.
Now let's discover the general formula for any polygon. We can divide any polygon into triangles by drawing lines from one vertex to all non-adjacent vertices. For a pentagon with 5 sides, we get 3 triangles. Since each triangle has angles summing to 180 degrees, the total is 3 times 180, which equals 540 degrees. The general formula is: sum equals n minus 2, times 180 degrees.