Question

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

Answer 1

Let ABCD be a quadrilateral.
Join A and C.

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

For $$\begin{gathered} △ABC: \\ \angle 2+\angle 4+\angle B={180}^o...\left(1\right) \end{gathered}$$

For $$\begin{gathered} △ADC: \\ \angle 1+\angle 3+\angle D={180}^o...\left(2\right) \end{gathered}$$

Adding (1) and (2):
$\left(\angle 1+\angle 2+\angle 3+\angle 4\right)+\angle B+\angle D={360}^o$

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

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