# Angle Sum Property of a Polygon

We will learn about the interior angle sum property of a polygon.

The sum of interior angles of a polygon can also be obtained without using the angle sum formula.

 In the adjoining figure of a triangle the sum of interior angles of a triangle is always 180°. i.e. for ∆ABC, ∠BAC + ∠ABC + ∠ACB = 180° i.e. ∠A + ∠B + ∠C = 180°
 In the adjoining figure of a quadrilateral ABCD, if diagonal BD of the quadrilateral is drawn, the quadrilateral will be divided into two triangles i.e. ∆ABD and ∆BDC. Since, the sum of interior angles of a triangle is 180°. Therefore, in ∆ABD, ∠ABD+ ∠BDA + ∠DAB = 180° and, in ∆BDC, ∠BDC + ∠DCB + ∠CBD = 180° 

∠ABD+ ∠BDA + ∠DAB + ∠BDC + ∠DCB + ∠CBD = 180° + 180°

⇒ (∠ABD + ∠CBD) + ∠DAB + (∠BDC + ∠BDA) + ∠DCB = 360°

⇒ ∠ABC + ∠DAB + ∠ADC + ∠DCB = 360°

⇒ ∠B + ∠A + ∠D + ∠C = 360°

 In the adjoining figure of a pentagon ABCDE, on joining AC and AD of the pentagon is divided into three triangles ∆ABC, ∆ACD and ∆ADE. Since, the sum of the interior angles of the triangles is 180° Therefore, the sum of interior angles of the pentagon ABCDE = Sum of interior angles of (∆ABC + ∆ACD + ∆ADE) = 180° + 180° + 180° = 540°
 In the adjoining figure of a hexagon ABCDE, on joining AC, AD and AE, the given hexagon is divided into four triangles i.e. ∆ABC, ∆ACD, ∆ADE and ∆AEF. The sum of the interior angles of the hexagon ABCDEF = sum of the interior angles of (∆ABC + ∆ACD + ∆ADE + ∆AEF) = 180° + 180° + 180° + 180° = 720°

Polygons

Polygon and its Classification

Terms Related to Polygons

Interior and Exterior of the Polygon

Convex and Concave Polygons

Regular and Irregular Polygon

Number of Triangles Contained in a Polygon

Angle Sum Property of a Polygon

Sum of the Exterior Angles of a Polygon