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CHAPTER-8
QUADRILATERALS
(CLASS IX)
Some Important Definitions and Key Points :
1.
Four sides closed figure is called quadrilaterals.
2.
Sum of the angles of a quadrilateral is 3600 .
3.
A diagonal of a parallelogram divides it into two congruent triangles.
4.
In a parallelogram,
5.
(a)
Opposite angles are equal
(b)
opposite sides are equal.
(c)
diagonals are not equal
(d)
sum of adjacent or co-interior angles is 1800.
(e)
opposite sides are parallel.
(f)
Diagonals bisects each other not at 900
A quadrilateral is a parallelogram, if
(a)
opposite sides are equal or
(b)
opposite angles are equal or
(c)
diagonals bisect each other or
(d)
opposite sides are parallel
(e)
a pair of opposite sides is equal and parallel
6.
Diagonals of a rectangle bisect each other and are equal and vice-versa.
7.
Diagonals of a rhombus bisect each other at right angles and vice-versa.
8.
Diagonals of a square bisect each other at right angles and are equal, and vice-versa.
9.
The line-segment joining the mid-points of any two sides of a triangle is parallel to the third side and
is half of it.
10.
A line through the mid-point of a side of a triangle parallel to another side bisects the third side.
11.
The quadrilateral formed by joining the mid-points of the sides of a quadrilateral, in order, is a
parallelogram.
12.
A parallelogram is said to be a rectangle if one of its angle is 90 0.
13.
A rhombus is said to be a square if one of its angle is 90 0.
14.
A parallelogram is said to be a rhombus of its adjacent sides are equal.
15.
In a quadrilateral if one pair of opposite sides is parallel then it is called trapezium.
Theorems
1.
Sum of the angles of a quadrilateral is 3600.
2.
A Quadrilateral is a parallelogram, if opposite sides are equal.
3.
A quadrilateral is a parallelogram, is opposite angle are equal.
4.
A diagonal of a parallelogram divides it into two congruent triangles.
5.
In a parallelogram, opposite sides and angles are equal.
6.
If each pair of opposite sides of quadrilateral is equal, then it is a parallelogram.
7.
If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.
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QD- 1
CHAPTER-8
QUADRILATERALS
(CLASS IX)
8.
The diagonals of a parallelogram bisect each other.
9.
If the diagonals of a quadrilateral bisect each, then it is a parallelogram.
10.
A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.
11.
The line drawn through the mid-point of one side of a triangle, parallel to another side bisects the
third side.
12.
The line segment joining the mid-points of the two sides of a triangle is parallel to the third side and
is half of it.
1 – Mark Questions
1.
2.
In a quadrilateral three angles are in the ratio 3 : 2 : 1 and one of the angles is 60 0, then find the other
angles.
In figure, ABCD is a parallelogram. If B  100 , then find the value of A  C  .
D
C
1000
A
B
3.
In figure, ABCD is a quadrilateral. If AO and BO be the bisectors of A and B respectively, then
find the value of x.
4.
Find the value of x in the given figure,
5.
In the given figure, ABCD is rhombus, if DAB  70 , then find CDB .
D
C
O
70
B
A
6.
In figure, ABCD is a rhombus in which BCD  110 , then find the value of x  y  .
A
B
X
Y
110
D
C
7.
ABCD is rhombus such that ACB  400 then find ADC
8.
If the angles of a quadrilateral ABCD , taken in order, are in the ratio, 3 : 7 : 6 : 4, then what is the
shape of ABCD
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QD- 2
CHAPTER-8
QUADRILATERALS
(CLASS IX)
9.
In a quadrilateral ABCD the angles A, B, C and D are in the ratio 3 : 4 : 5 : 6. then find the
difference between the greatest and the smallest angle.
10.
Under what condition a quadrilateral ABCD is said to be a parallelogram.
11.
In a square ABCD , the diagonals AC and BD bisects at O . Then what is the shape of AOB
12.
ABCD is a rhombus. If ACB  300 , then find ADB
13.
If the diagonals of a quadrilateral bisect each other, then which type of the quadrilateral.
14.
The diagonal AC and BD of quadrilateral ABCD are equal and are perpendicular bisector of each
other then which type of quadrilateral ABCD
15.
The sides of a quadrilateral extended in order to form exterior angler. Then find the sum of these
exterior angle.
16.
ABCD is rhombus with ABC  400 . Then fin the measure of ACD
17.
What is the figure obtained by joining the mid-points of the sides of a rhombus, taken in order.
18.
The diagonals AC and BD of a parallelogram ABCD intersect each other at the point O , if
DAC  320 and AOB  720 then find DBC
19.
Three angles of a quadrilateral are 600, 700, 800. Then find the fourth angle.
2 – Marks Questions
1.
Two adjacent angles of a parallelogram are in the ratio 2 : 3. Find all the four angles of the
parallelogram.
2.
Show that the diagonals of a rhombus are perpendicular to each other.
D
C
O
B
A
3.
ABCD is a parallelogram and the line segment AX, CY bisect the angles A and C, respectively.
Show that AX||CY.
D
A
4.
X
Y
C
B
In a parallelogram ABCD, ∠D=1050, determine the measures of ∠A and ∠B.
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QD- 3
CHAPTER-8
5.
QUADRILATERALS
The sides AB and CD of a parallelogram ABCD are bisected at E and F. Prove that EBFD is a
parallelogram (FIG).
D
F
C
B
E
A
6.
(CLASS IX)
ABCD is a parallelogram. The angle bisectors of A and C intersect at O. Find the measure of
AOD .
D
C
2
O
1
B
A
7.
In figure, ABCD is a parallelogram in which X and Y are the mid-points of the sides DC and AB
respectively. Prove that AXCY is a parallelogram.
D
C
B
Y
A
8.
X
In the given figure, ABCD and PQRC are rectangles and Q is the mid-point of AC, then prove that
(i)
DP = PC
(ii)
D
P
PR 
1
AC
2
C
R
Q
A
B
9.
Show that each angel of a rectangle is a right angle.
10.
In a quadrilateral ABCD, AO and BO are the bisectors of A and B , respectively. Prove that
1
AOB  C  D  .
2
11.
Two opposite angles of a parallelogram are 3x  2 and 63  2 x  . Find all the angles of the
parallelogram.


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QD- 4
CHAPTER-8
12.
QUADRILATERALS
(CLASS IX)
ABCD is a rhombus with ABC  56 . Determine ACD .
C
D
O
A
B
13.
P is a point on the side BC of a triangle ABC such that AB = AP. Through A and C lines are drawn
parallel to BC and PA, respectively, so as to intersect at D as shown in figures. Show that ABCD is a
cyclic quadrilateral.
D
A
P
B
14.
Show that the line segment joining the mid-points of opposite sides of a parallelogram, divided it
into two equal parallelogram.
F
D
C
B
E
A
15.
C
Show that the area of a rhombus is half the product of the lengths of its diagonals.
D
C
O
B
A
16.
In figure, ABCD is a parallelogram and E is the mid-point of AD. DL||BE meets AB produced at F.
Prove that B is the mid-point of AF and EB = LF.
C
D
L
E
F
B
A
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QD- 5
CHAPTER-8
QUADRILATERALS
(CLASS IX)
3 – Marks Questions
1.
The angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral.
2.
ABCD is a trapezium in which side AB is parallel to side DC and E is the mid-point of side AD. If F
is a point on the side BC such that the segment EF is parallel to side DC. Prove that F is the mid1
point of BC and EF   AB  DC  .
2
A
B
E
F
G
C
D
3.
P is a point on the side BC of a triangle ABC such that AB=AP. Through A and C lines are drawn
parallel to BC and PA respectively, so as to intersect at D as shown in figure. Show that ABCD is a
cyclic quadrilateral.
D
A
B
4.
C
P
In figure, ABCD is a trapezium in which AB||DC. E is the mid-point of AD and F is a point on BC
such that EF||DC. Prove that F is the mid-point of BC.
C
D
F
E
B
A
5.
ABCD is a rectangle in which diagonal AC bisects A and C . Prove that ABCD is a square.
6.
In figure, ABCD is a quadrilateral in which P, Q, R and S are the mid-points of the sides AB, BC,
CD and DA respectively. Show that PQRS is a parallelogram.
D
R
C
Q
S
A
P
B
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QD- 6
CHAPTER-8
7.
QUADRILATERALS
(CLASS IX)
ABCD is quadrilateral in which AD = BC and DAB  CBA (fig). Prove that
(i)
ABD  BAC
(ii)
BD = AC
(iii)
ABD  BAC
D
A
1
2
B
C
8.
Show that the bisectors of angles of a parallelogram form a rectangle.
9.
If the diagonals of a parallelogram are equal, then show that it is a rectangle.
10.
In a triangle ABC median AD is produced to X and that AD = DX. Prove that ABXC is a
parallelogram.
11.
Show that the diagonals of rhombus are perpendicular to each other.
D
C
O
B
A
12.
Prove that a diagonal of a parallelogram divides it into congruent triangles.
D
C
B
A
13.
Show that each angles of a rectangle is a right angle.
D
C
A
14.
B
ABCD is a parallelogram. If E is the mid-point of BC and AE is the bisector of A , prove that
1
AB  AD .
2
D
C
E
A
B
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QD- 7
CHAPTER-8
15.
QUADRILATERALS
(CLASS IX)
In the given figure, ABCD is a parallelogram in which diagonals AC and BD intersects at O. A line
segment LM is drawn passing through O. Prove that LO = OM.
D
L
C
O
A
B
M
16.
Prove that the diagonals of a square are equal and perpendicular to each other.
17.
In a triangle ABC median AD is produced to X such that AD = DX. Prove that ABXC is a
parallelogram.
18.
ABCD is a parallelogram and E is the mid-pint of side BC.DE and AB on producing meet at F.
Prove that AF=2AB.
19.
Points A and B are on the same side of a line m, AD  m and BE  m and meet m at D and E
respectively. If C is the mid-point of AB. Prove that CD = CE.
4 – Marks Questions
1.
In figure, ABCD is a trapezium in which AB||CD and AD=BC. Show that:
(i)
 A=  B
(ii)
A
B
E
D
2.
△ABC≅△BAD.
C
In the figure, PQRS is a square. M is the mid-point of PQ and AB  RM . Prove that RA=RB.
S
R
A
P
Q
M
B
3.
Prove that the diagonals of a square are equal and perpendicular to each other.
4.
AB and CD are respectively the smallest and longest sides of quadrilateral ABCD (FIG). Show that
A  C and B  D .
D
A
B
C
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QD- 8
CHAPTER-8
5.
QUADRILATERALS
(CLASS IX)
ABCD is a quadrilateral in which L, M, N and O are the mid-points of the sides AB, BC, CD and
DA respectively as in the figure below. Show that
(i)
ON || AC and ON 
1
AC
2
(ii)
ON = LM
LMNO is a ||gm.
C
N
D
(iii)
M
O
A
B
L
6.
Show that the line segment joining the mid-points of the opposite sides of a quadrilateral bisect each
other.
7.
Prove that the diagonals of a square are equal and perpendicular to each other.
8.
Show that the quadrilateral formed by joining the mid-points of the consecutive sides of a rectangle
is a rhombus.
C
R
D
S
Q
A
B
P
9.
ABCD is a parallelogram and AP, CQ are perpendicular drawn from vertices A and C on diagonal
BD. Show that AP = CQ.
10.
ABCD is a rhombus. Show that the diagonal AC bisects angle A and C and the diagonal BD bisects
angle B as well as angle D.
D
C
B
A
11.
ABCD is a parallelogram. AP the bisector of A and CQ the bisector of C meet the opposite sides
in P and Q respectively. Prove that AP || CQ.
D
A
P
Q
C
B
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QD- 9
CHAPTER-8
12.
QUADRILATERALS
In figure, ABCD is a parallelogram. If AB = 2AD and P is the mid-point of AB, then find CPD .
D
C
A
13.
(CLASS IX)
B
P
In figure, ABCD is a trapezium in which AB||CD and AD = BC. Show that:
(i)
A  B
ABC  BAD
(ii)
A
B
D
E
C
14.
In the quadrilateral ABCD, CO and DO are the bisectors of C and D respectively. Prove that:
1
COD  A  B  .
2
15.
Two parallel lines l and m are intersected by a transversal p. Show that the quadrilateral formed by
the bisectors of interior angles is a rectangle.
16.
In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively . Show that
the line segments AF and CE trisect the diagonal BD.
17.
ABCD is a trapezium in which AB ||CD and AD = BC. Show that
(i)
18.
A  B
(ii)
ABC  BAD
In a parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ . Show
that:
(i)
APD  CQB
(ii)
AP = CQ
(iii)
AQB  CPD
(iv)
AQ = CP
(v)
APCQ is a parallelogram.
19.
Bisectors of B and D of quadrilateral ABCD meet CD and AB produced at P and Q
1
respectively. Prove that P  Q ABC  ADC  .
2
20.
ABCD is a parallelogram. If AB = 2AD and P is the mid-point of AB, then find CPD .
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QD- 10
CHAPTER-8
QUADRILATERALS
(CLASS IX)
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QD- 11