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Question

# What is the sum of the measures of the angles of a convex quadrilateral? Will this property hold if the quadrilateral is not convex? (Make a non-convex quadrilateral and try!)

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Solution

## Let ABCD be a convex quadrilateral. The diagonal AC divides the quadrilateral into two triangles. ∠A+∠B+∠C+∠D=∠1+∠6+∠5+∠4+∠3+∠2=(∠1+∠2+∠3)+(∠4+∠5+∠6)=180∘+180∘ (angle sum property of a triangle)=360∘ Hence, the sum of measures of the angles of a convex quadrilateral is 360∘. This property holds good even for a non-convex quadrilateral. Let ABCD be a non – convex quadrilateral. The diagonal BD divides the quadrilateral into two triangles. Using angle sum property of triangle, In ΔABD, ∠1+∠2+∠3=180∘ ...(i)In ΔBDC, ∠4+∠5+∠6=180∘ ...(ii).Adding eq.(i) and (ii), we get,∠1+∠2+∠3+∠4+∠5+∠6=360∘⇒∠1+∠6+(∠3+∠4)+(∠2+∠5)=360∘ Hence proved.

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