In this paper, the properties of tangential and cyclic polygons proposed by Lopez-Real are proved rigorously using the theory of circulant matrices. In particular, the concepts of slippable tangential polygons and conformable cyclic polygons are defined. It is shown that an n-sided tangential (or cyclic) polygon P[subscript n] with n even is slippable (or conformable) and the sum of a set of non-adjacent sides (or interior angles) of P[subscript n] satisfies certain equalities. On the other hand, for a tangential (or cyclic) polygon P[subscript n] with n odd, it is rigid and the sum of a set of non-adjacent sides (or interior angles) of P[subscript n] satisfies certain inequalities. These inequalities give a definite answer to the question raised by Lopez-Real concerning the alternating sum of interior angles of a cyclic polygon. (Contains 5 figures.)