What do angles in a quadrilateral add up to

The sum of exterior angles (at vertices) add up to 360 degrees for any polygon (regular or not).  For example, an equilateral triangle has an exterior angle of 120 degrees at each vertex, which when multiplied by the number of vertices (3) makes 360 degrees.  The interior angle is (180-120)=60, so the sum of INterior angles is 3*(180)-360=(3-2)*180=180 degrees.

The same applies to a quadrilateral.
The exterior angles sum to 360 degrees, the sum of interior angles of a quadrilateral is therefore 4*(180)-360=(4-2)*180=360 degrees.

In general, for an n-sided polygon, the sum of interior angles is
(n-2)*180 degrees.

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