# Ex.3.1 Q3 Understanding Quadrilaterals Solution - NCERT Maths Class 8

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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!)

What is known?

Quadrilateral $$ABCD$$

What is unknown?

Sum of the measures of the angles of a convex quadrilateral.

Reasoning:

Let $$ABCD$$ be a convex quadrilateral. Then, we draw a diagonal $$AC$$ which divides the Quadrilateral into two triangles. We know that the sum of the angles of a triangle is $$180$$ degree, so by calculating the sum of the angles of a $$∆ABC$$ and , we can measure the sum of angles of convex quadrilateral.

## Text Solution

$$ABCD$$ is a convex quadrilateral made of two triangles $$∆ABC$$ and $$∆ADC.$$ We know that the sum of the angles of a triangle is $$180$$ degree. So:

\begin{align}\angle {\rm{6 + }}\angle {\rm{5 + }}\angle 4 &= 180^\circ \left[ {{\text{sum of the angles of } \Delta ABC = 180^\circ }} \right]{\rm{}}\\\angle {\rm{1 + }}\angle 2{\rm{ + }}\angle 3 & = 180^\circ \left[ {{\text{sum of the angles of }\Delta ADC = 180^\circ }} \right]\end{align}

\begin{align}\angle 6 &+ \angle 5 + \angle 4 + \angle 1 + \angle 2 + \angle 3\\ &= {{180}^{\rm{o}}} + {{180}^{\rm{\circ}}}\\ & = {{360}^{\rm{\circ}}}\end{align}

Hence, the sum of measures of the triangles of a convex quadrilateral is $$360^\circ$$. Yes, even if quadrilateral is not convex then, this property applies. Let $$ABCD$$ be a non-convex quadrilateral; join $$BD$$, which also divides the quadrilateral in two triangles

Using the angle sum property of triangle, again$$ABCD$$ is a concave quadrilateral, made of two triangles $$\Delta \text{ABD}$$ and $$\Delta \text{BCD}$$ .Therefore, the sum of all the interior angles of this quadrilateral will also be,

$180^\circ + 180^\circ=360^\circ$

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