Question

Prove that the sum of the angles of a quadrilateral is $360°$.

Answer 1

Step 1 : Assumption

Let us assume a quadrilateral $ABCD$ such that $AC$ is a diagonal which divides the quadrilateral into two triangles $ABC$ and $ADC$ as shown in the figure.

Step 2 : Finding the sum of the angles of a quadrilateral

By angle sum property of a triangle,

Sum of the interior angles of a triangle is $180°$

In $△ABC$

$\angle ABC+\angle BCA+\angle BAC=180°$ $...\left(1\right)$

In $△ADC$

$\angle ADC+\angle DCA+\angle DAC=180°$ $...\left(2\right)$

Adding $\left(1\right)$ and $\left(2\right)$, we get

$\angle ABC+\angle BCA+\angle BAC+\angle ADC+\angle DCA+\angle DAC=180°+180°$

$\angle ABC+\left(\angle BCA+\angle DCA\right)+\left(\angle BAC+\angle DAC\right)+\angle ADC=360°$

$\angle ABC+\angle BCD+\angle BAD+\angle ADC=360°$

Hence, it is proved that the sum of the angles of a quadrilateral is $\mathbf{360}\mathbf{°}$.