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In quadrilateral ABCD, AB, BC, CD and DA are the four sides A,B,C
the four vertices and A. B, C and D are the four angles formed a Now join the opposite vertices A to C and B to D [see Fig. 8.2
AC and BD are the two diagonals of the quadrilateral ABCD
In this chapter, we will study more about different types of quadrilaterals
properties and especially those of parallelograms
You may wonder why should we study about quadrilaterals for parallelog Look around you and you will find so many objects which are of the she quadrilateral the floss, walls, ceiling, windows of your classroom, the each face of the duster, each page of your book, the top of your study table se of these are given below (see Fig. 8.3).
Blackboard
Book
Table
Fig 4.3
Although most of the objects we see around are of the shape of special quadres called rectangle, we shall study more about quadrilaterals and especially parallelg because a rectangle is also a parallelogram and all properties of a parallelogrami true for a rectangle as well.
8.2 Angle Sum Property of a Quadrilateral
Let us now recall the angle sum property of a
quadrilateral.
The sum of the angles of a quadrilateral is 360 This can be verified by drawing a diagonal and dProperties of a Parallelogram perform an activity.
Cut out a parallelogram from a sheet of paper et it along a diagonal (see Fig. 8.7). You obtain
riangles. What can you say about these
les? Place one triangle over the other. Turn one around. ecessary. What do
you
observe?
Observe that the two triangles are congruent to h other.
Fig. 8.7
Repeat this activity with some more parallelograms. Each time you will observe ach diagonal divides the parallelogram into two congruent triangles.
Let us now prove this result.
rem &.1: A diagonal of a parallelogram divides it into two congruent
gles
Let ABCD be a parallelogram and AC be a diagonal (see Fig. 8.8). Observe the diagonal AC divides parallelogram ABCD into two triangles, namely, A ABC ACDA. We need to prove that these triangles are congruent.
In A ABC and A CDA, note that BC || AD and AC is a transversal. ZBCA = 2 DAC (Pair of alternate angles)
ABDC and AC is a transversal. <BAC / DCA (Pair of alternate angles)= BC
There is yet another property of a parallelogram. Let us study the same. Draw a logram ABCD and draw both its diagonals intersecting at the point O Fig &10)
ere the lengths of OA, OB, OC and OD. do you observe? You will observe that
OA= OC and OB = OD.
Ois the mid-point of both the diagonals.
peat this activity with some more parallelograms. ch time you will find that O is the mid-point of both the diagonals. we have the following theorem:
8. The diagonals of a parallelogram ct each other.
Now, what would happen, if in a quadrilateral me diagonals bisect each other? Will it be a
allelogram? Indeed this is true. This result is the converse of the result of A
eorem 8.6. It is given below:
8.7: If the diagonals of a quadrilateral
Fig. 8.10
sect each other, then it is a parallelogram. You can reason out this result as follows:
Note that in Fig. 8.11, it is given that OA = OC
OB=OD.
A AOB = A COD
(Why?)
efore, 2 ABO= 4 CDO (Why?)
this, we get AB || CD
milarly,
BCAD
refore ABCD is a parallelogram.
ly it can be shown that ZAPB-90 or 2 SPQ-90 ( it was shown for DSA Similarly, POR-90 and SRO 90"
PORS is a quadrilateral in which all angles are right angles
Can we conclude that it is a rectangle? Let us examine. We have shown that PSR-2 PQR90 and 2 SPQ-2 SRQ-90, So both pairs of opposite angles Therefore, PQRS is a parallelogram in which one angle (in fact all angles) is 90 and
PORS is a rectangle
Another Condition for a Quadrilateral to be a Parallelogram have studied many properties of a parallelogram in this chapter and you have also fed that if in a quadrilateral any one of those properties is satisfied, then it becomes parallelogram.
We now study yet another condition which is the least required condition for a
lateral to be a parallelogram.
It is stated in the form of a theorem as given below:
A quadrilateral is a parallelogram if a pair of opposite sides is
and parallel.
Look at Fig 8.17 in which AB = CD and
ABICD. Let us draw a diagonal AC. You can show
AABCA CDA by SAS congruence rule.
So BCAD (Why?)
Let us now take an example to apply this property fa parallelogram.
gle ABCD is a parallelogram in which P de mid-points of opposite sides AB and CD gee Fig. 8.18). If AQ intersects DP at S and BQ
snow take some examples.
Fig. 8.11
ple 1: Show that each angle of a rectangle is a right angle Let us recall what a rectangle is
ctangle is a parallelogram in which one angle is a right angle.
converse result? You whate Thus, Theorem
Each t
bisect
N
paralla
ABCD parallelogram? the a
Y
parallelogram, opposite angles converse also true? Yes.
quadrilateral, each pair parallelogram. Exar
Theor
twee Fi
What d
AC = CA
(Common)Fy observe that is the mid point of / passing through E and is parallel to C
Pe that AFCF by using the congruence of
All and & CDE
In & ABC, D, E and F are respectively
mid points of sides AB, BC and CA Fg827). Show that A ABC is divided into four went triangles by joining D. E and E
As D and E are mid-points of sides AB BC of the triangle ABC. by Theorem 8.9,
DE AC
DF || BC and EF || AB
edore ADEF, BDFE and DFCE are all parallelograms.
DE is a diagonal of the parallelogram BDFE.
ABDE A FED
iarty
A DAF = A FED
Pg 426)
MEF
find
WF[BC
Repeat this activity some So rive
The 2.5 The segment joining mid-points of t
191
a parallelogram ABCD, E and F are the ad points of sides AB and CD respectively ee Fig. 313 Show that the line segments AF EC wisect the diagonal BD
Show that the line segments joining the mid-points of the opposite sides of drilateral bisect each other.
ARC is a triangle right angled at C. A line through the mid-point M of hypoten AB
Fig. 8.31
and parallel to BC intersects AC at D. Show that Dis the mid-point of AC
(0) MDLAC
CM MA-AB
Summary
chapter, you have studied the following points: Sam of the angles of a quadrilateral is 360°.
A diagonal of a parallelogram divides it into two congruent triangles.
kaparallelogram.
opposite sides are equal
(i) opposite angles are equal
diagonals bisect each other Atrilateral is a parallelogram, if
opposite sides are equal
) diagonals bisect each other
or
(i) opposite angles are equal
a pair of opposite sides is equal and parallel Sapnals of a rectangle bisect each other and are equal and vice-versa.
Digonals of a rhombus bisect each other at right angles and vice versa Dagonals of a square bisect each other at right angles and are equal, and vice-versa
The line-segment joining the mid-points of any two sides of a triangle is parallel to the
Bedside and is half of it. Aline through the mid-point of a side of a triangle parallel to another side bisects the third
The quadrilateral formed by joining the mid-points of the sides of a quadrilateral, in order
parallelogram
You can prove theorem using following
Observe 8.25 in which and mid-points and respectively
& CDF
SLFDF and BEAEDC (Why?
The BCDE is parallelogram (Why?)
This gives BC
In this
Can you
that cove the theorem
below
The 8.18 The line drawn through the mid point of bisects the side
A EFCA FED
all the four triangles are congruent.
m and are three parallel lines
ted by transversals p and q such that I, m
P
on off equal intercepts AB and BC on p
Fig. 8.28). Show that 1, m and n cut off equal
pts DE and EF on q also.
We are given that AB - BC and have ve that DE EF.
join A to F intersecting mat G perium ACFD is divided into two triangles,
4 ABC A CDA
(ASA rule)
dagonal AC divides parallelogram ABCD into two congruent ges ABC and CDA. Now, sides of parallelogram ABCD. What do you observe?
measure the opposite You will find that AB = DC and AD = BC.
This is another property of a parallelogram stated below:
8.2 In a parallelogram, opposite sides are equal. You have already proved that a diagonal divides the parallelogram into two congruentividing the quadrilateral into two triangles,
Let ABCD be a quadrilateral and AC be a diagonal (see Fig. 8.4). What is the sum of angles in A A7DCY
Form
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