Question

write a proof of the polygon interior angle - sum theorem. the sum of the measures of the interior angles of a convex n - gon is ( 180 cdot (n - 2) ). by drawing every diagonal from one vertex in a convex, n - sided polygon, the polygon can be decomposed into triangles.

Explanation:

Step1: Understand the Polygon

Consider a convex \( n \)-gon. Let's pick one vertex. From this vertex, we draw diagonals to non - adjacent vertices.

Step2: Count the Triangles

When we draw diagonals from one vertex of a convex \( n \)-gon, the number of triangles formed inside the polygon is \( n - 2 \). For example, in a triangle (\( n = 3 \)), we can't draw a diagonal from a vertex to non - adjacent vertices, and the number of triangles is \( 3-2=1 \) (the triangle itself). In a quadrilateral (\( n = 4 \)), drawing a diagonal from one vertex divides it into \( 4 - 2=2 \) triangles.

Step3: Sum of Angles in Triangles

We know that the sum of the interior angles of a triangle is \( 180^{\circ} \). Since the \( n \)-gon is divided into \( n - 2 \) triangles, the sum of the interior angles of the \( n \)-gon is the sum of the interior angles of these \( n - 2 \) triangles.

Step4: Calculate the Sum

If each triangle has an interior angle sum of \( 180^{\circ} \) and there are \( n - 2 \) triangles, then the sum of the interior angles of the convex \( n \)-gon is \( 180^{\circ}\times(n - 2) \).

Answer:

To prove the Polygon Interior Angle - Sum Theorem (the sum of the interior angles of a convex \( n \)-gon is \( 180^{\circ}\cdot(n - 2) \)):

1. Select a Vertex in the Convex \( n \)-gon: Choose one vertex of the convex \( n \)-gon.
2. Draw Diagonals from the Vertex: From this chosen vertex, draw diagonals to all non - adjacent vertices. This action divides the convex \( n \)-gon into \( n - 2 \) non - overlapping triangles. For instance, a pentagon (\( n=5 \)) will be divided into \( 5 - 2 = 3 \) triangles when diagonals are drawn from one vertex.
3. Recall the Triangle Angle - Sum Property: The sum of the interior angles of a single triangle is \( 180^{\circ} \).
4. Calculate the Sum of Interior Angles of the \( n \)-gon: Since the \( n \)-gon is composed of \( n - 2 \) triangles, the sum of the interior angles of the \( n \)-gon is the sum of the interior angles of these \( n - 2 \) triangles. Mathematically, this is \( 180^{\circ}\times(n - 2) \), because if each of the \( n - 2 \) triangles has an interior angle sum of \( 180^{\circ} \), then the total sum is \( 180^{\circ}\cdot(n - 2) \).