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

      

Let ABCD be a quadrilateral.

Join A and C.

Now, we know that the sum of the angles of a triangle is 180°.

For $\triangle A B C$.

$\angle 2+\angle 4+\angle B=180^{\circ} \quad \ldots$ (1)

For $\triangle A D C:$

$\angle 1+\angle 3+\angle D=180^{\circ} \quad \cdots(2)$

Adding (1) and (2):

$(\angle 1+\angle 2+\angle 3+\angle 4)+\angle B+\angle D=360^{\circ}$

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

Hence, the sum of all the angles of a quadrilateral is 360°.