This lesson introduces more details than basics about quadrilaterals and covers
Special Quadrilaterals. It helps you on identification and learn properties of
special quadrilaterals. You’ll do it further from your earlier learning about polygons. You know that quadrilaterals may be divided into two major groups
i.e. parallelograms and other quadrilaterals. In addition to quadrilaterals
like the parallelogram, rectangle etc. others are classified as special
quadrilaterals. The properties and related details about special quadrilaterals,
you’ll understand in the contents presented by the instructor in own handwriting,
using video
and with the help of several examples with solution.
(More text below video...)
(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.
Given: Quadrilateral ABCD
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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||
and 
not parallel to 
Calculate the slopes now (Figure below):
