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A self-intersecting quadrilateral is called variously a cross-quadrilateral, crossed quadrilateral, butterfly quadrilateral or bow-tie quadrilateral. In a crossed quadrilateral, the four "interior" angles on either side of the crossing (two acute and two reflex, all on the left or all on the right as the figure is traced out) add up to 720°. [10]
It divides the quadrilateral into two congruent triangles that are mirror images of each other. [7] One diagonal bisects both of the angles at its two ends. [7] Kite quadrilaterals are named for the wind-blown, flying kites, which often have this shape [10] [11] and which are in turn named for a hovering bird and the sound it makes.
In general, any quadrilateral with perpendicular diagonals, one of which is a line of symmetry, is a kite. Every rhombus is a kite, and any quadrilateral that is both a kite and parallelogram is a rhombus. A rhombus is a tangential quadrilateral. [10] That is, it has an inscribed circle that is tangent to all four sides. A rhombus.
Because it is a regular polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. [6] Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the following isoperimetric inequality holds:
Each has an axis of symmetry parallel to and equidistant from a pair of opposite sides, and another which is the perpendicular bisector of those sides, but, in the case of the crossed rectangle, the first axis is not an axis of symmetry for either side that it bisects. Quadrilaterals with two axes of symmetry, each through a pair of opposite ...
A quadrilateral is equidiagonal if and only if [5]: p.19, [4]: Cor.4 =. This is a direct consequence of the fact that the area of a convex quadrilateral is twice the area of its Varignon parallelogram and that the diagonals in this parallelogram are the bimedians of the quadrilateral.
A Saccheri quadrilateral is a quadrilateral with two sides of equal length, both perpendicular to a side called the base. The other two angles of a Saccheri quadrilateral are called the summit angles and they have equal measure. The summit angles of a Saccheri quadrilateral are acute if the geometry is hyperbolic, right angles if the geometry ...
The orange and green quadrilaterals are congruent; the blue one is not congruent to them. Congruence between the orange and green ones is established in that side BC corresponds to (in this case of congruence, equals in length) JK, CD corresponds to KL, DA corresponds to LI, and AB corresponds to IJ, while angle ∠C corresponds to (equals) angle ∠K, ∠D corresponds to ∠L, ∠A ...