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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
Therefore, sum of interior angles of a non convex quadrilateral is also 360.
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