Angle Sum Property in Quadrilaterals

We have seen that in any triangle, the sum of the three angles is 180⁰. In any quadrilateral, the sum of the four angles is 360⁰. The following figure shows an example:

This fact will hold true even if one of the angles of the quadrilaterals is reflex, as shown below:

The justification of the angle sum property is based on the angle sum property of triangles.

Proof: Consider any quadrilateral ABCD, and draw the diagonal AC, as shown below:

The angles of triangle ABC sum to 180⁰, and the angles of triangle ADC also sum to 180⁰. Thus, the total sum of these angles (which is actually equal to the sum of the four angles of the quadrilateral) is 360⁰. Formally:

In \(\Delta ABC\), we have:

\(\angle ABC\) + \(\angle BCA\) + \(\angle CAB\) = 180⁰

Similarly, in \(\Delta ADC\), we have:

\(\angle ACD\)+ \(\angle CDA\)+ \(\angle DAC\) = 180⁰

Summing the two relations, we have:

\(\angle ABC\) + \(\angle CDA\) + (\(\angle BCA\) + \(\angle ACD\)) + (\(\angle CAB\) +\(\angle DAC\)) = 180⁰

è \(\angle ABC\) + \(\angle CDA\) + \(\angle BCD\) + \(\angle DAB\) = 360⁰

è\(\angle B\) + \(\angle D\) + \(\angle C\) + \(\angle A\) = 360⁰