Interior angles of an inscribed (cyclic) quadrilateral

Definition: Opposite pairs of interior angles of an inscribed (cyclic) quadrilateral are supplementary (add to 180 °).
Try this Drag any orange dot. Note how the red and blue pairs of angles are supplementary.

Recall that an inscribed (or 'cyclic') quadrilateral is one where the four vertices all lie on a circle. It turns out that the interior angles of such a figure have a special relationship. Each pair of opposite interior angles are supplementary - that is, they always add up to 180°.

In the figure above, drag any vertex around the circle. Note how the blue and red pairs of angles always add to 180°.

'Crossed' polygons

In the figure above, if you drag a point past its neighbor the quadrilateral will become 'crossed' where one side crossed over another. In such 'crossed' quadrilaterals the interior angle property no longer holds. (Most properties of polygons are invalid when the polygon is crossed). The angles instead become congruent (equal in measure).

Things to try

In the figure above

  1. Click on 'Hide details'.
  2. Drag the vertices around to create a new (uncrossed) quadrilateral.
  3. Calculate the measures of the two missing angles.
  4. Click "show details" to check your answer.

Other polygon topics


Types of polygon

Area of various polygon types

Perimeter of various polygon types

Angles associated with polygons

Named polygons