Question

The sum of angles of a quadrilateral is:

• A. ${180}^{\circ }$
• B. ${360}^{\circ }$
• C. ${270}^{\circ }$
• D. None of these

Answer 1

The correct option is B ${360}^{\circ }$

The sum of all the angles of a quadrilateral is ${360}^{\circ }$.

Consider a quadrilateral $ABCD,$

Then, $\angle A+\angle B+\angle C+\angle D={360}^{\circ }$.

Join vertices $A$ and $C,$ we get the diagonal $AC$.

Now, in $△ABC,$

$\angle CAB+\angle ABC+\angle BCA={180}^{\circ }[$Sum of all angles of a triangle is ${180}^{\circ }]$.....$\left(1\right)$

In $△ACD,$

$\angle CAD+\angle ADC+\angle DCA={180}^{\circ }[$Sum of all angle of a triangle is ${180}^{\circ }]$....$\left(2\right)$

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

$(\angle CAB+\angle ABC+\angle BCA)+(\angle CAD+\angle ADC+\angle DCA)={180}^{\circ }+{180}^{\circ }$

$⟹$ $\angle ABC+\angle ADC+(CAB+CAD)+(BCA+DCA)={360}^{\circ }$

$⟹$$\angle ABC+\angle ADC+\angle BAD+\angle BCD={360}^{\circ }$

$\therefore \angle A+\angle B+\angle C+\angle D={360}^{\circ }$

That is, the sum of all the angles of a quadrilateral is ${360}^{\circ }$