This is a free lesson from our course in Geometry Other useful lessons:
(Continued from above) A quadrilateral is a polygon with four sides, and special quadrilaterals are a unique class of quadrilaterals which have at least one pair of sides parallel. E.g. say a rectangle. Though the quadrilaterals may be of different types, still they have common features like:
� all the quadrilaterals have four sides and are coplanar.
� each quadrilateral has two diagonals, and sum of the four interior angles of a quadrilateral adds to 360�. However, note that there are differences too, which makes the quadrilaterals different.
Further it explains about the five major special quadrilaterals and their properties. E.g. parallelogram in which both pairs of opposite sides are parallel to each other. Notice that the opposite sides and opposite angles of the parallelogram are congruent, and also the diagonals bisect each other.
A rectangle is a quadrilateral type where all interior angles are right angles. More over it has all the properties of a parallelogram, added by that its diagonals are congruent.
A rhombus is a type of parallelogram in which all the sides congruent. It has all the properties of a parallelogram, and the diagonals are perpendicular as well.
A square is a rectangle whose sides are all congruent. This has all the properties of a parallelogram and a rhombus.
Trapezoids have different properties; say it has only one set of sides parallel to each other, 0pposed to two in other cases above.
Such unique properties shall help you to identify the special quadrilaterals and in turn help you in solving the problems in relation to discovering unknown features of figures.
For example: prove that quadrilateral ABCD is an isosceles trapezoid.
Proof: You can do it by by proving || and not parallel to Calculate the slopes now (Figure below):  = 2/2  = 1 = 5/5  = 1 = -4/1 = -4 = -1/4  = -0.25
It is known that the slopes of parallel lines are equal, therefore, AB || DC. The slopes of AD and BC are unequal; therefore, these are not parallel. Also the legs of the trapezoid are congruent, this trapezoid is isosceles. Hence proved.
Remember:
Important Theorem Statements:
Theorem 1: the diagonals of a parallelogram bisect each other.
Theorem 2: if the diagonals of a quadrilateral bisect each other then the quadrilateral is a parallelogram
Theorem 3: a quadrilateral is a parallelogram if one pair of opposite sides are equal and parallel
Theorem 4: the diagonals of a rectangle are equal in length
Theorem 5: the diagonals of a rhombus are perpendicular to each other.
Theorem 6: the diagonals of a square are equal and perpendicular to each other
Notice that if you apply the theorems of parallelograms in relation to a triangle, it can be concluded: �The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and half in length�.
The video above will explain more in detail about Special Quadrilaterals and their relationship, with the help of several examples.
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