CHAPTER – 1
UNIT – 4
SETS
EXERCISE – 1.4.3
1. Define a set. Give examples to illustrate the difference between a
collection and a set
Definition: A set is a collection of well defined objects. The objects of a set
are called elements or members of the set.
Examples:
a) The collection set of all prime numbers between 100 and 200.
b) The collection of all planets in the universe.
c) The collection of all fair people in the city
Here (a) and (b) are examples of sets. But (c) is not one cannot define fair.
2. Which of the following collection are sets?
(a) All the students of your school.
(b) Members of Indian parliament.
(c) The colures of rainbow.
(d) The people of Karnataka having green ration card.
(e) Good teachers in a school
(f) Honest persons of your village.
Ans: (a), (b) and (c) are sets.
(d), (e) and (f) are not sets.
3. Represent the following sets in roster method:
(a) Set of all alphabet in English language.
(b) Set of all odd positive integers less than 25.
(c) The set of all odd integers.
(d) The set of all rational numbers divisible by 5.
(e) The set of all colors in the Indian flag.
(f) The set of letters in the word ELEPHANT.
Ans: (a) A = {a, b, c……..x, y, z}
(b) Z = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23}
(c) P = {±1, ±3, ±5….}
(d) R = {5, 10, 15…...}
(e) Y = {saffron, white, green}
(f) S = {E, L, P, H, A, N, T}
4. Represent the following sets by using their standard notations.
(a) Set of natural numbers
(b) Set of integers
(c) Set of positive integers
(d) Set of rational numbers
(e) Set of real numbers
Ans: (a)
N = {1, 2, 3……}
(b)
Z = {0, ±1, ±2, ±3…….}
(c)
Z + = {1, 2, 3…….}
(d)
Q = {p/q, p, q  z and q ≠ 0}
(e)
R = (Q  Z}
5. Write the following sets in set builder form:
(a) {1,4,9,16,25,36}
(b) {2, 3, 5, 7, 11, 13, 17, 19, 23……}
(c) {4, 8, 12, 16, 20, 24……}
(d) {1, 4, 7, 10, 13, 16……}
Ans: (a) A = {x: /x=k2 for some kN, 1  k  6}
(b) P = {x/x is a prime number}
(c) X = {x/x is a multiple of 4}
(d) Z = {x/x=3n – 2 when n = 1, 2, 3…..}
6.
State whether the set is finite or infinite:
(a) The set of all prime numbers.
(b) The set of all sand grains on this earth.
(c) The set of points on a line.
(d) The set of all school in this world.
Ans: (a) infinite
(b) finite
(c) Infinite
(d) finite
7. Check whether the sets A and B are disjoint
(a) A is the set of all even positive integers. B is the set of all prime
numbers.
(b) A = {3, 6, 9, 12, 15……}
B = {19, 24, 29, 34, 39……..}
(c) A is the set of all perfect squares; B is the set of all negative integers.
(d) A = {1, 2, 3} and B = {4, 5,{1, 2, 3}}
(e) A is the set of all hydrogen atoms in this universe; B is the set of all
water molecules on earth.
Ans: (a) A and B are not disjoint sets since A  B = {2}
(b) A and B are not disjoint since A  B = {24....}
(c) A and B are disjoint.
(d) A  B = {1, 2, 3}. Hence they are not disjoint.
(e) A and B is disjoint.
EXERCISE 1.4.4
1. Find union of A and B and represent it using venn diagram.
(i)
A = {1, 2, 3, 4, 8, 9}, B = {1, 2, 3, 5}
(ii)
A = {1, 2, 3, 4, 5}, B = {4, 5, 7, 9}
(iii)
A = {1,2,3}, B = {4, 5, 6}
(iv)
A = {1, 2, 3, ,4 ,5}, B = {1, 3, 5}
(v)
A = {a, b, c, d}, B = {b, d, e, f}
Ans: (i) A  B = {1, 2, 3, 4, 5, 8, 9}
A
4
1
8
9
2
3
5
B
(ii)
A  B = {1, 2, 3, 4, 5, 7, 9}
A
1
2
3
(iii)
4
5
9
A  B = {1, 2, 3, 4, 5, 6}
A
B
4
1
2
5
3
(iv)
B
7
6
A  B = {1, 2, 3, 4, 5}
A
2
1
4
3
5
B
A  B = {a, b, c, d, e, f}
(v)
A
a
b
e
c
d
f
B
2. Find the intersection of A and B, and respect it by Venn diagram:
(i)
A = {a, c, d, e}, B = {b, d, e, f}
(ii)
A = {1, 2, 4, 5}, B = {2, 5, 7, 9}
(iii)
A = {1, 3, 5, 7}, B = {2, 5, 7, 10, 12}
(iv)
A = {1, 2, 3}, B = {5, 4, 7}
(v)
A = {a, b, c}, B = {1, 2, 9}
Ans: (i) A  B = {d, e}
A
a
b
c
d
e
f
B
(ii) A  B = {2, 5}
A
1
7
B
2
4
5
9
(iii) A  B = {5, 7}
A
1
2
3
(iv) A  B = {
A
5
10
7
12
B
}
1
2
3
5
4
7
B
(v) A  B = {
}
A
a
B
1
b
2
c
9
3. Find A  B and A  B when:
(i)
A is the set of all prime numbers and B is the set of all
composite natural numbers:
(ii)
A is the set of all positive real numbers and B is the set of all
negative real numbers:
(iii)
A = N and B = Z:
(iv)
A = {x /x  Z and x is divisible by 6} and
B = {x / x  Z and x is divisible by 15}
(v)
A is the set of all points in the plane with integer coordinate
and B is the set of all points with rational coordinates.
Ans:
(i)
A = {2, 3, 5, 7….}
B = {1, 4, 6….}
A  B = {1, 2, 3, 4} = N
AB={
(ii)
}
A = R+
B = R+
A  B = R – {10}
i.e. A  B ={set of non zero real numbers}
AB={
(iii)
}
A=N B=N
AB=Z
AB=N
(iv)
A  B = {x / x  Z and x is divisible by 6 and 15} and
A  B = {x / x  Z and x is divisible by 30}
[LCM of 6 and 15 = 30]
(v)
A  B = the set of all points with rational co-ordinates = B.
A  B = the set of all points with rational co-ordinates = A.
4. Give examples to show that
(i)
A  A = A and A  A = A
(ii)
If A  B, then A  B = B and A  B =A. can you prove these
statements formally?
Ans: (i) If A = {2 4 6 8}
Then A  A = {2, 4, 6, 8……} = A
A  A = {2, 4, 6, 8…….} = A
Hence A  A = A and A  A = A
(ii) A = {1, 3, 5, 7, 9……}
B = {1, 2, 3, 4, 5……}
We see that A  B
A  B = {1 2 3 4…..} = B
A  B = B
A  B = {1, 3, 5……} = A
A  B = A
5. What is A   and A   for a set A?
Ans: A   = A
A   =
EXERCISE 1.4.5
1. If A' = {1, 2, 3, 4}, U = {1, 2, 3, 4, 5, 6, 7, 8}, find A in U and draw Venn
diagram
Ans: A' = {5, 6, 7, 8}
U
A
1
5
2
3
4
6
7
8
2. If U = {x/x  25, xN}. A = {x/x  U, x  15} and B = {x/x  U, 0  x 
25}, list the elements of the following sets and draw Venn diagram:
(i)
A' in U:
(ii)
B' in U
(iii)
A\B;
(iv)
AΔB
Ans: U = {1, 2, 3, 4 ……….25}
A = {1, 2, 3, 4……….15}
B = {1, 2, 3 ….25}
(i)
A' = {16, 17, 18, 19….25}
U
23
16
12345
6 7 8 9 10
11 12 13
14 15
17
20
(ii)
18
22
B' = {
10
24
21 25
}
20 12 21
3 22 13 5 2
14 16 29 19
6 7 17 24 9
15 25 8 11
10 18
(iii)
A
A\B = {
B
A'
}
A
B
2 13 16 25
14 1 3 5 17 22
6 12 15 18
21
9 8 11
19
23
10 20 24
A\B
(iv)
AΔB=A\BB\A
={
}  {16, 17, 18... 25}
= {16, 17, 18 …..25}
A
B
3 2 1 16
456
17 18
7 8 9 10 19 20
11 12 13 21 22
14 15 23
24 25
AΔB
3. Let A and B subsets of a set U. Identify the wrong statements:
(i)
(A')' = A
(ii)
A\B =B \A
(iii)
A  A' = U
(iv)
AΔB=B ΔA
(v)
(A \ B)' = A' \ B'
Ans: If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8, 9}
(i)
(A')' = {2, 4, 6, 8}
(A')' = {1, 3, 5, 7, 9} = A
 (A')' = A
(ii)
A \ B = {1, 3, 5, 7}
B \ A = {2, 4, 6, 8}
We see that A \ B ≠ B \ A
(iii)
A  A' = {1, 3, 5, 7, 9} U {2, 4, 6, 8}
= {1, 2, 3, 4, 5, 6, 7, 8, 9}
=U
 A  A' = U
(iv)
A Δ B =(A \ B) U (B \ A)
= {1, 3, 5, 7} U {2, 4, 6, 8}
= {1, 2, 3, 4, 5, 6, 7, 8}
B Δ A = (B \ A) U (A \ B)
= {2, 4, 6, 8} U {1, 3, 5, 7}
= {1, 2, 3, 4, 5, 6, 7, 8}
 AΔB=B ΔA
(v)
A \ B = {1, 3, 5, 7}
(A \ B)' = {2, 4, 6, 8, 9}
A' = {2, 4, 6, 8} and B' = {1, 3, 5, 7}
A' \ B' = {2, 4, 6, 8}
Hence (A \ B)' ≠ A' \ B'
4. Suppose U = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}. A = {3, 4, 5, 6, 9}, B = {3, 7,
9, 5} and C = {6, 8, 10, 12, 7}. Write down the following sets and draw
Venn diagram for each:
(i)
A'
(iv) (A')'
(ii)
B'
(iii)
(v)
(B')' (iv)
C'
(C')'
Ans: (i) A' = {7, 8, 10, 11, 12, 13}
U
7
A
11
3
8
4
12
5
6 9
10
13
(ii) B' = {4, 6, 8, 10, 11, 12, 13}
U
B
4
6
5
7
9
8
10
3
11
12
13
(iii) C' = {3, 4, 5, 9, 11, 13}
U
C
3
6
4
7
8
5
10
13
12
9
11
(iv) A' = {7, 8, 10, 11, 12, 13}
(A')' = {3, 4, 5, 6, 9} = A
U
A
13
7
3
11
5
8
6 9
10
4
12
(v) (B')' = B' = {4, 6, 8, 10, 11, 12, 13}
(B')' = {3, 7, 9, 5} = B
U
B
4
13
3
6
7
8
5
11
9
12
10
(vi) (C')' = {3, 4, 5, 9, 11, 13}
(C')' = {6, 8, 10, 12, 7} = C
U
3
4
C
13
6
5
8
10
9
12
7
11
13
5. Suppose U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4} and B = {2, 4, 6, 8},
write down the following sets and draw Venn diagram.
(i)
A'
(ii)
B'
(iii)
(iv)
AB
(v)
(A  B)'
AB
(vi)
(A  B)'
How (A  B)' is related to A' and B'? What relation you see between
(A  B)' and A' and B'
Ans: (i) A' = {5, 6, 7, 8, 9}
U
A
5
1
7
8
2
3
4
6
9
(ii)
B' = {1, 3, 5, 7, 9}
U
B
1
6
3
(iii)
7
4
9
8
A  B = {1, 2, 3, 4, 6, 8}
U
(iv)
5
2
A
B
1
2
6
3
4
8
A  B = {2, 4}
U
A
B
1
2
6
3
4
8
(v)
(A  B)'
(A  B) = {1, 2, 3, 4, 6, 8}
(A  B)' = {5, 7, 9}
A
U
5
B
1
2
6
3
4
8
9
7
(vi)
(A  B)'
(A  B) = {2, 4}
(A  B)' = {1, 3, 5, 6, 7, 8}
U
A
5
B
1
2
6
3
4
8
7
We see that (A  B)' = A'  B'
(A  B)' = A'  B'
9
6. Find (A \ B) and (B \ A) for the following sets and draw Venn diagram.
(i)
A = {a, b, c, d, e, f, g, h} and
B = {a, e, i, o, u}
(ii)
A = {1, 2, 3, 4, 5, 6} and
B = {2, 3, 5, 7, 9}
(iii)
A = {1, 4, 9, 16, 25} and
B = {1, 2, 3, 4, 5, 6, 7, 8, 9}
(iv)
A = {x | x is a prime number less than 5} and
B = {x | x is a square number less than 16}
Ans:
(i) A = { a, b, c, d, e, f, g, h}
B = {a, e, i, o, u}
A \ B = {b, c, d, f, g, h}
A
B
b f
c g
d
h
a
e
i
o
u
B \ A = {i, o, u}
A
B
b
c d
f g
h
a
e
i
o
u
(ii) A = {1, 2, 3, 4, 5, 6} and
B = {2, 3, 5, 7, 9}
A \ B = {1, 4, 6}
A
B
1
4
2
3
5
7
9
6
B \ A = {7, 9}
A
B
1
4
6
2
3
5
7
9
(iii) A = {1, 4, 9, 16, 25} and
B = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A \ B = {16, 25}
A
B
16
25
1
4
2
3
6
8
5
9 7
B \ A = {2, 3, 5, 6, 7, 8}
A
B
16
25
1
4
2
3
6
8
5
9 7
(iv) A = {x | x is a prime number less than 5}
= {2, 3}
B = {x | x is a square number less than 16}
= {1, 4, 9}
A \ B = {2, 3}
A
B
2
1
4
3
9
B \ A = {1, 4, 9}
A
B
2
1
4
3
9
7. Looking at the Venn diagram list the elements of the following gsets:
(i)
A\B
(ii)
B\A
(iii)
A\C
(iv)
C\A
(v)
B \C
(vi)
C \B
A
B
1
2
3
5
11
13
7
6
8
9
12
C
Ans:
(i)
A \ B = {1, 2, 7}
(ii)
B \ A = {5, 6}
(iii)
A \ C = {1, 2, 3}
(iv)
C \ A = {6, 8, 9}
(v)
B \ C = {5, 3}
(vi)
C \ B = {7, 8, 9}
8. Find A Δ B and draw Venn diagram when:
(i)
A = {a, b, c, d} and B = {d, e, f}
(ii)
A = {1, 2, 3, 4, 5} and B = {2, 4}
(iii)
A ={1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6}
(iv)
A = {1, 4, 7, 8} and B = {4, 8, 6, 9}
(v)
A = {a, b, c, d, e} and B = {1, 3, 5, 7}
(vi)
A = {1, 2, 3, 4, 5} and B = {1, 3, 5, 7}
(i)
A = {a, b, c, d}
Ans:
B = {d, e, f}
A \ B = {a, b, c}
B \ A = {e, f}
A Δ B = {a, b, c, e, f}
A
B
a
e
d
b
c
(ii)
A = {1, 2, 3, 4, 5}
A \ B = {1, 3, 5}
B\A={ }
A Δ B = {1, 3, 5}
f
B = {2, 4}
A
B
1
3
5
(iii)
2
4
A ={1, 2, 3, 4, 5}
B = {1, 2, 3, 4, 5, 6}
A \ B = {.}
B \ A = {6}
A Δ B = {6}
A
B
1
2
3
5
(iv)
A = {1, 4, 7, 8}
B = {4, 8, 6, 9}
A \ B = {1, 7]
B \ A = {6, 9}
A Δ B = {1, 6, 7, 9}
A
B
1
7
4
6
8
9
(v)
A = {a, b, c, d, e} and B = {1, 3, 5, 7}
A \ B = {b, d}
B \ A = {g}
A Δ B = {b, d, g}
A
B
b
d
(vi)
a
c
e
g
A = {1, 2, 3, 4, 5} and B = {1, 3, 5, 7}
A \ B = {2, 4}
B \ A = {7}
A Δ B = {2, 4, 7}
A
B
2
4
1
3
5
7
ADDITIONALPROBLEMS ON SETS
1. Which of the following collections of objects are sets?
(i)
All the months in a year.
(ii)
All the short boys of your class.
(iii)
The entire whole numbers less than 50.
(iv)
All the poor numbers of Indian.
Ans: (i) and (ii)
2. Rewrite the following statements using the sets notations:
(i)
p is an element of set A.
(ii)
a and b are members of set C.
(iii)
a is an empty set and B is a non empty set.
(iv)
Q does not belong to set B,
(v)
All elements of A which are not in B.
(vi)
Symmetric difference of A and B.
(i)
PA
(ii)
a, b  c
(iii)
A = { } and B ≠ { }
(iv)
QB
(v)
A\B
(vi)
AΔB
Ans:
3. Represent the following sets in roster form:
(i)
A = {x | x  N, x  7}
(ii)
B = {x | x is a factor of 32}
(iii)
C = {x | x = 2n, n  N , n  5}
(iv)
D = {x | x is a letter of the word CARELESS}
(v)
E = {x | x is a two digit number the sum of digits being 8}
Ans:
(i)
A = {1, 2, 3, 4, 5, 6, 7}
(ii)
B = {±1, ±2, ±4, ±8, ±16, ±32}
(iii)
C = {2, 4, 6, 8, 10]
(iv)
D = {C, A, R, E, L, S}
(v)
E = {17, 71, 80, 62, 26, 53, 35, 46}
4. Represent the following sets in set builder form:
(i)
A = {5 6 7 8 9 10 11 12}
(ii)
B = {1 2 3 4 6 8 12 16 24 48}
(iii)
C = {11 13 17 19 23 29 31 37}
(iv)
E = {Atlantic, Arctic}
Ans:
(i)
A = {x | x N and 5  x  12}
(ii)
B = {x | x is a factor of 48}
(iii)
C = {x | x is a prime number between 10 and 38}
(iv)
E = {x | x is polar regions of earth}
5. State whether the following statements are true or false.
(i)
The empty set  = {0}
(ii)
The empty set has no subset.
(iii)
If A = {x | 3 < x  10, r N}, then 3  A.
(iv)
The collection of interesting drams written by Shakespeare is a well
defined set.
Ans:
(i)
False. Empty set is denoted as { } or .
(ii)
False.
(iii)
False. A = {4, 5, 6, …….10}
(iv)
False. The word interesting cannot be defined.
6. Write the following statements in words:
(i)
P  Q = {a e i o u}
(ii)
R  S =
(iii)
If A  B, then A  C.
(iv)
If x  A and A  B, then x  B.
(v)
If x  P, then x  P'
(i)
The union of sets P and Q is the set of vowels.
(ii)
The intersection of set R and S is a null set.
(iii)
If A is a subset of B then A is a subset of B, then x belongs to B.
(iv)
If x belongs to A and A is a subset of B, then x belongs to B.
(v)
If x belongs to the set P, then x does not belongs to the complement of
Ans:
set P.
7. In the adjacent diagram write the elements of
(i)
A
(ii)
B
(iii)
AB
(iv)
AB
(v)
A\B
(vi)
B\A
(vii) A Δ B
A
a
b
d
f
e
g
c
h
Ans:
(i)
A = {a, b, c, d, e}
(ii)
B = {f, g, h, d, e}
(iii)
A  B = {a, b, c, d, e, f, g, h}
(iv)
A  B = {d ,e}
(v)
A \ B = {a, b, c}
(vi)
B \ A = {f, g, h}
(vii) A Δ B = (A \ B)  (b \ A)
= {a, b, c}  {f, g, h}
= {a, b, c, f, g, h}
B
8. In the adjacent figure – express the following sets in terms of A and B.
(i)
{1, 2, 3}
(ii)
{1, 2, 3, 4, 5, 6, 7}
(iii)
{2, 3, 5, 6, 7}
(iv)
{2, 3}
(v)
{1, 4}
(vi)
{1, 4, 5, 6, 7}
A
B
5
1
2
4
3
2
6
7
Ans:
(i)
{1, 2, 3} = A
(ii)
{1, 2, 3, 4, 5, 6, 7} = A  B
(iii)
{2, 3, 5, 6, 7} = B
(iv)
{2, 3} = A  B
(v)
{1, 4} = A \ B
(vi)
{1, 4, 5, 6, 7} = A Δ B
9. Using set symbols, write down expressions for the shaded portion in the
following Venn diagram.
A
B
A
B
A
B
Ans:
(i)
A\B
(ii)
B\A
(iii)
(A  B)'
10. If U is the set of all letters in English alphabet and if A = {a b c d},
B ={b c d e}, draw Venn diagrams to represent.
(i)
AB
(ii)
A'  B'
(iii)
A\B
(iv)
A\ A  B
(v)
AΔB
Ans:
(i)
A  B = {a, b, c, d, e}
U
f
s
A
t
g
e
a
r
B
r u
b
c
z
t
q
e
e
d
I
jk l
xv
mn
o
p
w
(ii)
A'  B'
A' = {e, f, g …… x, y, z}
B' = {a, f, g…….x, y, z}
A'  B' = {f, g, h ……x, y, z}
U
f
s
A
t
e
a
g
B
z
r u
b
c
e
r
t
q
e
d
I
jk l
(iii)
xv
mn
o
A
e
p
w
A \ B = {a}
f
s
t
g
a
r
B
r u
b
c
z
t
q
e
e
d
I
jk l
xv
mn
o
p
w
(iv)
A\ A  B
A  B = {b c d}
A\ A  B = {a}
U
f
s
A
t
e
a
g
r
B
z
r u
b
c
t
q
e
e
d
I
jk l
xv
mn
(v)
o
p
w
AΔB
A \ B = {a}
B \ A = {e}
A Δ B = (A \ B)  (B \ A)
U
f
s
A
t
g
e
a
r
B
r u
b
c
z
t
q
e
e
d
I
jk l
xv
mn
o
p
w
NON-TEXTUAL QUESTIONS
1. Which of the following are sets?
(i)
The collection of stars.
(ii)
The collection of all smart students in a class.
(iii)
The collection of engineers in India
(iv)
The collection of beautiful women in Karnataka.
Ans:
(i)
And (iii) are sets.
But (ii) the word smart cannot be defined hence it is not a set.
(iv) The word beautiful cannot be defined, since is not a set.
2. Match each of the set on the left in the roster form with the same set on
the right described in set-builder form.
(i)
{1, 2, 3, 6} (a) {x : x is a prime number and a divisor of 6}
(ii)
{2, 3} (b) {x : x is an odd natural number less than 10}
(iii)
{M, A, T, H, E, I, C, S} (c) {x : x is a natural number and a divisor
of 6}
(iv)
{1, 3, 5, 7, 9} (d) {x : x is a letter of the word MATHEMATICS}
Solution:
The sets in (a): {2, 3} 
(ii)  (a)
The set in (b): {1, 3, 5, 7, 9}
(iv)  (b)
The set in (c): {1, 2, 3, 6}
(i)  (c)
The set in (d): {M,A,T,H,E,M,A,T,I,C,S } = { M, A, T, H, E, I,
C, S
 (iii)  (d)
3. Which of the following are examples of a null set
(i)
Set of odd natural numbers divisible by 2.
(ii)
Set of even prime numbers.
(iii)
{x : x is a natural number, x < 5 and x > 7}
(iv)
{x : x is an even prime number greater than 2}
Solution: (i), (iv) are null sets.
4. Looking at the Venn diagram write the elements of the following sets:
(i)
ABC
(ii)
(A  B) \ C
(iii)
(B  C)  A'
(iv)
(A  C)  B'
(v)
A'  B'  C
(vi)
(B \ A) \ C
A
B
5
2
6
3
4
7
C
Solution: A = {1, 2, 4, 5} B = {1, 2, 3, 6}
C = {1, 3, 4, 7}
U = {1, 2, 3, 4, 5, 6, 7, 8}
(i)
A  B  C = {2}
(ii)
(A  B) \ C
A  B = {1, 2}
(A  B) \ C = {2}
(B  C)  A'
(iii)
B  C = {1, 3} and A’ = {3, 6, 7, 8}
(B  C)  A' = {3}
(A  C)  B'
(iv)
A  C = {1, 4} and B = {4, 5, 7, 8}
(A  C)  B' = {4}
A'  B'  C
(v)
A' = {3, 6, 7, 8} B' = {4, 5, 7, 8}
A'  B' = {7, 8}
A'  B'  C = {7}
(vi)
(B \ A) \ C
B \A = {3, 6}
(B \ A) \ C = {6}
5. Find A Δ B and draw Venn diagram when
(i)
A = {2, 4, 6, 8} b = {2, 3, 5, 7}
Solution: A \ B = {4, 6, 8}
B \ A = {3 5 7}
(A \ B)  (B \ A) = {4, 6, 8}  {3 5 7}
= {3, 4, 5, 6, 7, 8}
A
B
4
6
3
2
5
8
7
(ii)
U = {a, b, c, d, e, f, g, h, i, j, k}
A = {a, d, f, g, k}
B = {c, f, h, i, j, k}
A \ B = {a, d, g}
B \ A = {c, h, i, j}
A Δ B = (A \ B)  (B \ A)
= {a, d, g}  {c, h, I, j}
= {a, c, d, g, h, i, j}
A
B
a
d
e
f
k
g
h i
j
b
e
6. Four alternatives are given for the following questions or statements.
Select the most appropriate answer:
a) The set theory was introduced by the mathematician
a) George Contor
c) Euclid
b) srinivasa Rmanujan
d) John Venn
(a)
b) A suitable example for set is …….
a) A group of good teachers of your school.
b) Collection of all tall people of Bangalore.
c) Group of all honest politicians.
d) The set of all students in your class who have scored 90% and above. (d)
c) Given set G = {2 3 7 8 9}, which of the following is a wrong statement?
a) 2  G
b) 7  G
c) 11  G
d) 9  G
(b)
d) From the set D = {d, s, e, r, t}, the maximum number of subsets we can
form is …….
a) 5
b) 10
c) 25
d) 32
(d)
e) The shaded portion of this Venn diagram represents
a) A  B
b)A \ B
c) B \ A
d) A  B
(b)
f) For the set {1, 3, 4, 6}, which of the following given sets can be
considered a universal set?
a) {1, 2, 3, 4, 5}
b) {3, 5, 6, 7}
c) {1, 2, 3, 4, 5, 6, 7}
d) {1, 5, 6, 7}
(c)
g) The set of all points common to any two distinct parallel lines is an
example for………
a) Unit set
b) Null set
c) Finite set
d) Infinite set
(b)
h) The method of representing the intersection of two sets is
a) A  B = {x | x  A and x  B}
b) A  B = {x | x  A or x  B}
c) A \ B = {x | x  A and x  B}
d) B \ A = {x | x  A and x  B}
(a)
i) Given A = {a b c d e}, B = {a e i o u}, their symmetric difference is ……..
a) {a, e}
b) {b, c ,d}
c) {b, c, d, i, o, u}
d) {i, o, u}
(c)
7. Fill in the blanks:
(i)
A set which contains no elements ……. (null set)
(ii)
Set of all the points that lies on a plane. This is an example for the set.
(Infinite set)
(iii)
X = {2, 3, 4, 7, 8, 9}, and X  Y = {3, 4, 7}, then X \Y = ……(2 8 9)
(iv)
If Q = {k, v, n}, and U = {a, b, c, k, x, p, u, n, m}, then (Q')' = ……..
(Q')' = {k, v, n} = Q
(v)
Representation of the set of all the perfect cubes from zero to 100 in
rule method is ……… (x | x = n3, 0  n3  100}
8. Match the following.
List A
1. Singleton
List B
a) Set of all perfect cubes
between 5 and 6
2. Finite set
b) Set of all prime numbers
3. Infinite set
c) Two sets without common
element
4. Null set
d) Sets of all counting numbers
from 1 to 100
5. Disjoint sets
e) Set of all the points common
to two intersecting lines
Ans:
(i)
1. e
(ii)
2. d
(iii)
3. b
(iv)
4. a
(v)
5. c
CHAPTER – 4
UNIT – 3
THEOREMS AND PROBLEMS ON
PARALLELOGRAMS
EXERCISE 4.3.2
1. Suppose ABCS is a parallelogram and the diagonals intersect at E. Let
PEQ be a line segment with P on AB and Q on CD. Prove that PE = EQ.
Solution:
Given ABCD is a parallelogram in which AC and BD intersect at E. The line
segment PEQ meet AB at P and CD at Q.
To prove: PE = EQ
D
Q
C
E
A
P
B
Proof:
Statement
In ΔAPE and
Reason
ΔCQE
Diagonals AC and
AE = CE
BD bisect each other.
EAP = CEQ
AP | | QC, AC
Transversal alternate angles.
AEP = CEQ
vetially opposite angles
 ΔAPE  ΔCQE
ASA
 PE = EQ
C.P.C.T
2. Let ABCD be a parallelogram. Let BP and DQ be perpendiculars respectively
from B and D on to AC. Prove that BP = CQ.
Solution:
D
C
P
Q
A
B
Proof:
Statement
Reason
In parallelogram
ABCD
Given
BP  AC  DO  AC
In ΔADQ and ΔCBP
opposite sides of
AD = CD
a parallelogram.
DAQ = BCP
AD | | BC, AC
Transversal, Alternate interior
Angles.
DQA = BPC = 90
Given
ΔADQ = ΔCBP
SAA
BP =DQ
C.P.C.T
3. Prove that in a rhombus, the diagonals are perpendicular to each other.
Solution:
A
B
E
D
C
Given: ABCD is a rhombus; AC and BD intersect at E.
To prove: AC  BD
Proof:
Statement
Reason
In rhombus, ABCD
AC and BD
Intersect each Other at E
Given
In ΔABE and ΔADE
sides of a rhombus.
AB = AD
Diagonals bisect
BE = DE
each other
AE common
 ΔABE = ΔADE
SSS
 AEB + AED
CPCT
Proof AEB + AED = 180 Linear pair
 AED = AED =
180
2
= 90
 AC and BD are perpendicular to each other.
4. Suppose in a quadrilateral the diagonals bisect each other
perpendicularly. Prove that the quadrilateral is a rhombus.
Solution:
D
C
E
A
B
Data: ABCD is a quadrilateral, DB  AC,
AE = EC, BE = ED
To prove: ABCD is a rhombus
Proof:
In Δ ADE and Δ DEC,
AE = EC
(data)
DE = ED
(common side)
AED = DEC
90
 Δ AED = Δ DEC (RHS)
 AD = DC
(CPCT)
Similarly we can prove
AD =AB
DC = BC
So we can conclude
AB = BC = DC = AD
 ABCD is a quadrilateral with equal sides and has perpendicularly
bisecting diagonals.
 ABCD is a Rhombus.
5. Let ABCD be a quadrilateral in which ΔABD  ΔBAC. Prove that ABCD
is a parallelogram.
Solution:
C
D
B
A
Data: ABCD is a Quadrilateral
ΔABD  ΔBAC
To prove: ΔABD 
BD = AC
(CPCT)
 Area of ΔABD = Area of ΔBAC
And both the triangles stand on same base AB.
 DC | | AB
But AD | | BC
 AD | | BC
Hence opposite sides are parallel ABCD is a parallelogram.
EXERCISE 4.3.3
1. The area of parallelogram is 153.6 cm2. The base measures 19.2 cm. what
is the measurement for the height of the parallelogram?
Solution:
D
C
h
A
B
Area of the parallelogram = bh = 153 cm2
Base = b = 19.2cm
19.2 × h = 153.6
h=
153 .6
19.2
=
1536
192
= 8cm
2. In parallelogram ABCD, AD = 25cm and AB = 50 CM. if the altitude from a
vertex D on to AB measure 22 cm, what is the altitude from a vertex B on to
AD?
Solution:
D
C
22
25
A
h
E
B
Area of parallelogram = bh
Area ABCD = 50 × 22cm2
If the altitude from B to AD is h,
And ABCD = 25 × h cm2
25 x h = 50 × 22
h =
15 ×22
25
= 44cm
3. In a parallelogram ABCD, AB = 4x and AD = 2x + 1. If the perimeter is
38cm and area 60cm2 find the length of the altitude from D on to AB.
Solution:
D
A
C
E
B
Perimeter = 2 (l + b) = 38
l+ b =
38
2
= 19
4x + 2x + 1 = 19
6x = 19 – 1 = 18
x=
18
6
=3
AB = 4x = 4 × 3 = 12cm
Let the altitude DE = h
Area = bh = 12 × h = 60
h=
60
12
= 5cm.
4. Let ABCD be a parallelogram and consider its diagonal AC. Draw
perpendiculars BK and DL to AC.
Prove that BK = DL.
Solution:
D
C
K
L
A
B
Given: ABCD is a parallelogram. BK and DL are perpendiculars drawn
To diagonals AC.
Proof:
Statement
Reason
In quadrilateral.
ABCD DL  AC, BK  AC
given
ΔADC  ΔABC
Diagonal divides a
Parallelogram into
Two congeuent
Triangles
 Area Δ ABC = Area Δ ADC
i. e.
1
2
1
AC × BK = AC × DL
2
BK = DL
5. Let ABCD be a parallelogram. Prove that 2 area (ABCD)  AC × BD.
Solution:
Given: ABCD is a parallelogram
D
C
G
F
E
A
B
To prove: 2 areas (ABCD)  AC × BD.
Construction: Draw DF and BG perpendicular to AC
Proof:
Area ABCD
= area ΔABC + area ΔADC
1
1
2
2
= AC × DF + AC × BG
1
= AC (DF + BG)
2
2 area (ABCD)
1
= 2 × AC (DF + BG)
2
= AC (DF + BG)
In right ΔDFE, DF < hypotimise DE
Similarly in ΔBGE, BG < BE
Thus 2 area ABCD = AC (DF + BG)
 AC (DE + BE)
i.e. 2 areas ABCD  AC × BD
2 areas ABCD will be equal to AC × BD if AC  BD AC will be
perpendicular to BD if ABCD is a rhombus or a square.
 2 area ABCD = AC × BD in a rhombus or in a square.
6. Prove that the area of triangles standing on the same base or equal bases
and between same parallels are equal in area.
Solution:
D
E
A
l
C
B
m
Given: ABCD and ΔABF stand on the same base and are between the same
Parallel l and m
To prove: area ΔABD = area ΔABF
Construction: drew parallelograms ABCD and ABFE
Proof:
Statement
Reason
1. Area of parallelogram ABCD
And ABFE are equal
They stand on the same base
and are between the same
Parallels.
2. Area ΔABD =
1
2
area ΔABCD
Area of triangles is equal to
Half the area of a parallelogram
Stand base and between the
Same parallels.
3. Similarly area
1
ΔABF = area ABFE
2
4. Area ΔABD = area ΔABF from (1), (2) and (3)
EXERCISE 4.3.4
1. Prove the statement of example 4 using the following ideas: Join DF and
product it to meet AB produced in G. Show the two parallelograms
ABFE and EFCD have equal area using the converse of mid-point
theorem.
Solution:
D
C
E
F
G
A
B
Data: ABCD is a parallelogram E and F is the mid-points of AD and BC
Respectively.
To prove: Area of | | gm ABFE = area of | | gm FEDC
Proof: In Δ DCF and Δ BFG,
DCF = FBG
(alternate angles, DC | | AB)
DFC = BFG
(V.O.A)
FC = FB
(F is midpoint)
 Δ DCF  Δ BFC (ASA postulate)
 DC = BG
But
DC = AB
 AB = BG
Now area of | | gm ABEF = 2 area of BGF
(1)
(∵Parallelogram and Δ standing on equal base and between same parallel)
Now Area of | | gm EDCF = 2 area of Δ DCF
But Δ DCF  Δ BGFS
 Area of | | gm EDCF = 2 area of Δ BGF (2)
From (1) and (2)
Area of | | gm ABEF = Area of | | gm EDCF
2. Prove the converse of the mid-point theorem following the guidelines
given below.
Consider a triangle ABC with D as the min-point of AB. Draw DE | | BC
to intersect AC in E. let E 1 be the mid-point of ac. Use mid-point
theorem to get DE 1 | | BC and DE1 =
𝐁𝐂
𝟐
. Conclude E = E 1 and hence E is
the mid-point of AC.
Solution:
A
E1
D
B
E
C
Given: in Δ ABC, D is the mid-point of AB. DE | | BC
To prove: E is the midpoint of AC
Proof:
Statement
1. In Δ ABC, D is the mid-point
Reason
Given
Of AB. DE | | BC
Let E, be the mid-point of AC.
Join DE, D is the mid-point of
AB and E1 is the mid-point of AC
 DE1 | | BC
Mid-point theorem parallels.
But DE | | BC
Given
This is possible only if E and E1 consider (Through a given point, only
one line can be drawn | | to be a given line)
 E and E1 coincide
i.e. DE1 is the same as DE
Thus a line drawn through the mid-point of a side of a triangle and
parallel to another bisects the third side.
3. In a rectangle ABCD, P, Q, R and S are the mid-point of the sides AB,
BC, CD and DA respectively. Find the area of PQRS in terms of area of
ABCD.
Solution:
D
R
C
S
Q
A
P
B
Given: ABCD is a triangle, P, Q, R and S are the midpoints of AB, BC, CD and
DA respectively. PQ, QR, RS and SP are joined.
To find: the areas (PQRS) in terms of area (ABCD)
Solution: S and Q are the mid-point of AD and BC
 SQ divides rectangles ABCD into two rectangles equal in area.
1
Area (ABQS) = area (SQCD) = area (ABCD)
2
Δ SRQ and parallelogram (rectangles) SQCD stand on the same base SQ and are
between the same parallel.
1
 Area Δ SRQ = area (SQCD)
2
1
Similarly area Δ SPQ = area (SABQ)
2
1
1
2
2
Adding, area Δ SRQ + area Δ SPQ = area (SQCD) +
1
i.e., area (PQRS) = area (ABCD)
2
area (SABQ)
4.
Suppose x, y and z are the mid-point of the sides PQ, QR and RP
Respectively of a triangle PQR. Prove that XYPZ is a parallelogram.
Solution:
P
X
Q
Z
Y
R
Given: x, y and z are the mid-point of the sides PQ, QR and RP of Δ PQR, xy and
Xz are joined.
Prove: XYRZ is a parallelogram.
Proof:
Statement
Reason
In PQR, x, y, and z are the
Mid-point of PQ QR and RP
Respectively.
XY | | PR
XY =
1
2
PR
1
mid-point theorem
mid-point theorem
ZR = PR
Given
 XY RZ is a
one pair of opposite sides
Parallelogram
parallel and equal.
2
5. Suppose EFGH is a trapezium in which EF is parallel to HG. Through
X, the mid-point EH, XY is drawn parallel to EF meeting FG at Y.
prove that XY bisects FG.
Solution:
H
X
G
P
E
Y
F
Given: EFGH is a trapezium with EF | | HG. Through X the mid-point of EH,
Is drawn parallel to EF
To prove: XY bisects FG
Construction: Join HF to intersect XY at P
Proof
Statement
In Δ HEF XP | |
 XP bisects HF PY | | HG
Reason
converse of mid-point theorem
HY | | EF and EF | | HG
Δ HGF, PY passes through
Mid-point of HF and is
Parallel to HG
 PY bisects FG
converse of mid-point theorem
6. Prove that if the mid-points of the opposite sides of a quadrilateral are
joined, they bisect each other.
Solution:
D
R
C
S
A
Q
P
B
Data: ABCD is a quadrilateral P, Q, R and S are mid-points of AB, BC, CD and
DA respectively.
To prove: PR and SQ bisect each other.
Construction: join AC.
Proof:
Statement
Reason
In Δ ABC, PQ | | AC, and
mid-point
1
PQ = AC
theorem ……. (1)
In Δ ADC, SR | | AC, and
mid-point
2
1
SR = AC
theorem ……. (2)
 PQ | | SR and PQ = SR
from (1) and (2) opposite sides
 PQRS is a parallelogram
equal and parallel
2
PR and SQ are diagonals
PR bisect SQ
EXERCISE 4.3.5
1. Construct parallelograms ABCD with the following measurements:
(a) AD = 4.2 cm, DC = 5.8 cm, D = 43
(b) AB = 43. Cm, BC = 3.2 CM, C= 120
(c) AD =4.2 cm, DC = 5.8 cm, AC = 3.4cm
(d) AD = 4.3 cm, DC = 5.7 cm, DB = 7cm
(e) AD = 6.8 cm , DB = 7.6 cm, AD = 6.4cm
Solution:
(a) AD = 4.2cm, DC = 5.8cm, D = 43
B
C
5.8
43
A
4.2
D
Steps of construction:
1. Draw a line segment AD = 4.2cm.
2. Draw DC = 5.8cm such that ADC = 43
3. With A as center and 5.8cm as radius draw another arc to intersect the above
at B.
4. Join AB and CB.
ABCD is the required parallelogram.
(b) AB = 4.3cm, BC = 3.2 cm, C= 120
E
D
C
120
3.2cm
A
4.3cm
B
Steps of construction:
C = 120  A =120 (opposite angles of a parallelogram)
BC = 3.2cm  A = 3.2cm (opposite angles of a parallelogram)
1. Draw a line segment AD = 3.2cm.
2. At A make an angle of 120 and draw AE.
3. In off AD = 3.2cm on AE.
4. With B as center and 3.2cm as radius draw an arc.
5. With D as center and 3.2cm as radius draw another arc to intersect the
above at C.
6. Join BC and DC.
ABCD is the required parallelogram.
(c) AD =4.2 cm, DC = 5.8 cm, AC = 3.4 cm
B
C
5.8
3.4
A
4.2
D
Step of construction:
1. Draw AD = 4.2cm
2. With A as center and 3.4 cm as radius draw an arc.
3. With D as center and 5.8cm as radius draw another arc to intersect the
above arc
4. Join AC and DC.
5. With A as center and 5.8 cm radius draw an arc.
6. With C as center and 4.2cm as radius draw another arc to intersect the
above at B.
7. Join AB and CB.
ABCD is the required parallelogram.
(d) AD = 4.3 cm, DC = 5.7 cm, DB = 7cm
AD = 4.3 cm DC = 5.7 cm = AB
Step of construction:
1. Draw AD = 5.7cm
2. With A as center and 4.3 cm as radius draw an arc.
3. With B as center and 7cm as radius, draw another arc to intersect the above
at D.
4. Join AD and BD
5. With B as center and 4.3 cm as radius draw an arc.
6. With D as center and 5.7 cm as radius draw another arc to intersect the
above at C.
7. Join DC and BC.
ABCD is the required parallelogram
(e) AD = 6.8cm , DB = 7.6cm, AD = 6.4cm
D
C
3.8
3.4
6.8
3.4
3.8
A
B
Diagonals AC and BD bisect each other at E
AE = EC =
AE = EC =
6.8
2
7.6
2
= 34
= 3.8
Step of construction:
1. Draw AD = 6.4cm.
2. With A as center and 3.4 cm as radius draw an arc.
3. With D as center and 3.8cm as radius draw another arc to intersect the
above arc at E.
4. Join AE and produce to C such that EC = 3.4cm.
5. Join DE and produces to B such that EB = 3.8cm.
6. Join AB, BC and DC.
ABCD is the required parallelogram
2. Cons trust rectangle ABCD with the following data:
(a) AB = 4cm, BC = 6cm
D
C
6cm
A
4cm
B
Step of construction:
1. Draw AB = 4cm.
2. Construct a perpendicular to AB at B and cut off BC = 6cm.
3. With A as center and 6cm as radius draw an arc.
4. With C as center and 4cm as radius draw another arc to intersect the
above arc at D.
5. Join AD and CD.
ABCD is the required parallelogram.
(b) AB = 6cm , AC = 7.2 cm
F
E
D
C
A
6
B
Step of construction:
1. Draw AB = 6cm.
2. At B construct ABE = 90.
3. With A as center and 7.2 cm as radius draw an arc to intersect the above
at C.
4. Join AC.
5. Construct BAF = 900.
6. With B as center and 7.2 cm as radius draw an arc to intersect the above
at D.
7. Join CD.
ABCD is the required parallelogram.
3. Construct square ABCD.
a) Which has side – length 2 cm
b) Which has diagonal 6cm
Solution:
a) Which has side – length 2 cm
E
D
A
C
2cm
B
Step of construction:
1. Draw AB = 2cm
2. At A draw AE  AB
3. With A as center and 2cm as radius draw an arc to intersect AE at D.
4. With D as center and 2cm as radius draw an arc.
5. With B as center and 2cm as radius draw another arc to intersect the
above at C.
6. Join BC and DC
ABCD is the required parallelogram.
b) Which has diagonal 6cm
D
3cm
A
6cm
C
3cm
B
Step of construction:
1. Draw AC = 6cm.
2. Draw the perpendicular bisecular of AC to intersect it at E.
3. With E as center and 3 cm as radius draw an arc to intersect the above at
B and D.
4. Join AB, BC, CD, and DA.
ABCD is the required parallelogram.
4. Construct rhombus ABCD such that:
a) AB = 3.2 cm, AC =4.8 cm
b) AB = 4.4 cm, BD = 5.4 cm
c) AC = 7 cm, BD = 4 cm
d) BD =6.8cm, AC = 5.4 cm.
Ans:
a) AB = 3.2 cm, AC =4.8 cm
D
C
4.8cm
A
B
Step of construction:
1. Draw AC = 3.2 cm.
2. With A as center and 4.8 cm as radius draw an arc.
3. With B as center and 3.2 cm as radius draw another arc to intersect to
above at C.
4. Join AC and BC.
5. With A as center and 3.2 cm as radius draw an arc.
6. With C as center and 3.2 cm as radius draw another arc to intersect the
above at D.
7. Join AD and CD.
ABCD is the required parallelogram.
b) AB = 4.4 cm, BD = 5.4 cm.
D
C
5.4cm
A
4.4cm
B
Step of construction:
1. Draw a AB = 4.4 cm.
2. With A as center and 4.4 cm as radius draw an arc.
3. With B as center and 5.4 cm as radius draw another arc to intersect to
above at D.
4. Join AD and BD.
5. With B as center and 4.4 cm radius draw an arc.
6. With D as center and 4.4 cm as radius draw another arc to intersect the
above at C.
7. Join DC and BC.
ABCD is the required parallelogram.
c) AC = 7cm, BD = 4 cm
P
D
A
7cm
C
4cm E
B
Q
Step of construction:
1. Draw AC = 7 cm.
2. Draw perpendicular bisector of AC to intersect or it at E.
3. Cut off ED = EB = 2 cm.
4. Join AB, BC, CD and DA.
d) BD = 6.8 cm, AC = 5.4 cm
P
A
B
6.8cm
D
E 5.4cm
C
Q
Step of construction:
1. Draw BD = 6.8 cm.
2. Draw PQ the perpendicular bisector of BD to intersect or it at E.
3. Cut off EA = EC = 2.7 cm.
4. Join AB, BC, CD and DA.
ABCD is the required square.
ADDITIONAL PROBLEMS
THEOREMS AND PROBLEMS ON
PARALLELOGRAMS
1. In the adjoining figure, D, E, F is the mid-point of the sides BC, CA and
AB respectively of triangle ABC. Show that
EDF = A,
DEF = B
and
DEF = C.
A
F
E
B
D
C
Given: D, E, F are the mid-points of sides BC, CA, AB of Δ ABC respectively.
DE < EF, and FD joined.
To prove: EDF = A, DEF = B and DFE = C
Proof:
Statement
Reason
D and E are the mid-point
Of BC and CA
 DE | | AB and DE =
Mid-point
1
2
Similarly, AE | | FD and
AB
theorem
AE = FD AFDE is a
opposite sides parallel
Parallelogram
and equal.
EDF = A
opposite angles of
Similarly DEF = B and
Parallelogram
DEF = CS
2. In the adjoining figure, ABCD is a parallelogram. E is the mid-point of
DC. Through D a line segment is drawn parallel to meet CB produced
at G and it cuts AB at F. prove that
(i) AD =
𝐆𝐂
𝟐
(ii) DG = 2EB.
Solution:
D
A
E
F
B
G
C
Given: ABCD is a parallelogram E is the mid-point of CD. EB is joined.
DG | | EB. DG meets CB produced at G. DG intersects AB at F.
To prove: (i) AD =
GC
2
(ii) DG =EB
Proof:
Statement
Reason
E is the mid-point of DC.
And DG | | EB
 In ΔCDG
Converse of mid-point
EB bisects CG
theorem
1
 CB = CG
2
But CB =AD
 AD =
1
2
CG
Opposite sides of a
parallelogram
E and B are the mid-points of
CD and CG of Δ CDG
 EB | | DG and
EB =
1
2
DG
 DG = 2EB.
Mid-point
theorem
3. In the adjoining figure, ABCD is a rhombus. A square DCFE is
constructed on the side CD. Join FB and AC. Let them intersect in G.
Prove that
AGB = 45.
Solution:
E
F
D
C
G
A
B
Given: ABCD is a rhombus = A square DCFE stands on DC.
To proof: ACB = 45
Proof: ABCD is a rhombus diagonal AC bisects BCD.
 BCA = DCA = x
DCF is a Square stands on a side DC
In Δ BCF, BC = CF
 BCF = CBF
 2 CFB + CBF + BCF = 180
CBF + CBF + (90 + 2x) = 180
= 2 CBF = 180 – 90 – 2x
2 CBF =
90 − 2x
2
 CBF =
90
2
–
2x
2
= 45 – x
Δ FGC,
CFG + FGC + CGF = 180
45 – x + FGC 90 +x = 180
FGC + 135 = 180
FGC
= 180 – 135
FGC
= 45
ABG = FGC vertically opposite angles
 AGB = 45
4. Let ABCD be a parallelogram in which E and F are the mid-points of
the sides AB and CD respectively. Prove that the line segments CE and
AF trisect the diagonals.
Solution:
D
F
C
Q
P
A
E
B
Given: In parallelogram ABCD, E and F are the mid-points of AB and DC
Respectively. AF and CF are joined. Diagonal BD is drawn.
To prove: CE and AF trisect BD.
Proof:
Statement
Reason
E and F are the mid-point
AB = CD opposite sides of a
Of AB and DC
parallelogram
 AE = FC, AE | | FC
 AB =
 AECF is a
one prior of opposite sides
Parallelogram
parallel and equal
1
1
2
2
CD
 AF | | EC
Δ BAQ, EP | | AQ and E is
The mid-point of AB
 EP, bisects BQ
 BP = PQ
Similarly from Δ DPC
QF | | PC and QF bisects DP.
 PQ =DQ
 BP = PQ = QD
i.e. EC and AD trisects BD.
converse of mid-point theorem
5. Prove that the line segment joining the mid-point of the diagonals of a
trapezium is parallel to the parallel sides and equal to half their
difference.
Solution:
D
C
P
X
M
N
A
B
Given: ABCD is a trapezium. Diagonals AC and BD intersect at P. AB | | DC.
M and N are the mid-points of AC and BD. MN is joined.
To prove: MN | | AB | | DC and XM – NX =
1
2
(AB – DC)
Construction: draw NX | | DC
Proof: In Δ BCD, N is the mid-point BD and NX | | DC
1
 NX = DC
2
[Converse of mid-point theorem]…….. (1)
Now draw a line parallel to AB from X, it will meet AC at M.
 XM and NM lie on the same line
XM =
1
2
AB
[Converse of mid-point theorem]…….. (2)
MN | | AB | | DC
From (1) and (2)
XM – NX =
1
2
AB –
1
2
DC
1
= (AB – DC)
2
6. Let ABCD be a parallelogram. Extend AD to E such that DC = DE. Join
EC and extend it to meet AB extended in F. Prove that BC = BF.
Solution:
E
D
A
C
B
F
Given: ABCD is a parallelogram. AD is a produced to E such that DC = DE. EC is
Joined.
To prove: BC = BF
Proof:
Statement
In Δ EDC, DE = DC
Reason
Given
 E = ECD = x (say)
Angles opposite
But E = BCF
to equal sides
CB | | AD (opposite sides of a
Parallelogram)
 CB | | EA, EF transversal
(Corresponding angles equal)
7. In the adjoining figure, ABCD is a parallelogram. Lines EF and GH are
drawn parallel to AB and BC respectively and they intersect in K prove
that the parallelogram HBFK and KGDC have equal area.
Solution:
D
G
E
C
F
K
A
H
B
Given: ABCD is a parallelogram
EF | | AB and GH | | BC
DEFC and HBCG are also parallelogram
To prove: Area of | | gm HBFX = Area of | | gm KCDE
Proof: Area (| | gm HKE) + Area (| | HBFK) =
A (| | gm AHCE + | | KCDE)
Since Δ gm KFGC is common
Area of | |gm HBFK = Area of | | gm KGDE
8. In a quadrilateral ABCD, the diagonals AC and BD intersect in E
suppose the area of the four triangles AEB, BEC, CED and DEA are
equal. Prove that ABCD is a parallelogram
Solution:
D
C
E
A
D
Given: In quadrilateral ABCD, diagonal AC and BD intersect at E. Area
Δ AEB = area Δ BEC = area Δ CED = area Δ DEA
To prove: ABCD is a parallelogram.
Proof:
Statement
Reason
Δ AEB and Δ BEC
Given
Stand on bases AE and CE
And have the same 3rd vertex B
Area Δ AEB = area Δ BEC
AE = CE
Δ0 standing on equal bases and
Having the same 3rd vertex are
Equal in area.
Similarly, BE = DE
 Ac and BD bisect each other
ABCD is a parallelogram
diagonals bisect each other