1. Parallelograms

2. Quadrilaterals A polygon with four sides is called a quadrilateral.B C D
A polygon with four sides is called a quadrilateral.
In a quadrilateral,
(a) any two sides that have a common vertex are called adjacent sides,
(b) any two non-adjacent sides are called opposite sides,
e.g. AB and BC are a pair of adjacent sides with
common vertex B,
e.g. AD and BC are a pair of opposite side without
a common vertex,

3. Quadrilaterals A polygon with four sides is called a quadrilateral.B C D
A polygon with four sides is called a quadrilateral.
In a quadrilateral,
(c) any two interior angles that are not lying on the same side are called opposite angles,
(d) the sum of all the interior angles is 360.
e.g. A and C are a pair of opposite angles,
sum of interior angles of a quadrilateral
= (4 – 2)  180

4. Properties of Parallelograms
A parallelogram is a quadrilateral that has two pairs of parallel opposite sides.
What are the properties of a parallelogram?

5. The opposite sides of a parallelogram are equal.B C D
Property I
AD = BC
The opposite sides of a parallelogram
are equal.
i.e. AB = DC,
[Abbreviation: opp. sides of // gram] A B C D
Property II
i.e. A = C,
B = D
The opposite angles of a parallelogram
are equal.
[Abbreviation: opp. s of // gram]

6. [Abbreviation: diags. of // gram]C D
Property III O
The diagonals of a parallelogram bisect
each other.
i.e. AO = OC,
BO = OD
[Abbreviation: diags. of // gram]

7. Can you find the unknowns in the figure?B C D
30 y
7 cm
x cm
In the figure, ABCD is a parallelogram. = AB DC ∴
(opp. side of // gram) = 7 x  = Ð +
180 C B ∵
(int. s, AB // DC)  = +
180 30 y ∴  =
150 y

8. Follow-up question Find the unknowns in the figure. Solution
4 cm
Find the unknowns in the figure. A B C D
2x – 15
b cm
(7 – b) cm
75
Solution
a cm
∵ BC = AD (opp. sides of // gram) 4 = a ∴
∵ AB = DC (opp. sides of // gram) 7 - = b ∴ 7 2 = b 5 . 3 = b

9. Follow-up question (cont’d)
4 cm
Find the unknowns in the figure. A B C D
2x – 15
b cm
(7 – b) cm
75
Solution
a cm
∵ B = D (opp. s of // gram) ∴  - = 15 2 75 x  = 90 2 x  = 45 x

10. Identifying Parallelograms
How can you determine whether a quadrilateral is a parallelogram?
Very good! Can you think of any other conditions to identify a parallelogram?
By the definition of a parallelogram, if the quadrilateral has two pairs of parallel opposite sides, it is a parallelogram.

11. Both pairs of opposite sides are equal. and AD = BC, i.e. if AB = DC
Condition I A B C D
Both pairs of opposite sides are equal.
and AD = BC,
i.e. if AB = DC
then ABCD is a parallelogram.
[Abbreviation: opp. sides equal ]
Condition II A B C D
Both pairs of opposite angles are equal.
i.e. if A = C
and B = D,
then ABCD is a parallelogram.
[Abbreviation: opp. s equal ]

12. The diagonals bisect each other. i.e. if AO = OC and BO = OD,
Condition III
The diagonals bisect each other. O
i.e. if AO = OC
and BO = OD,
then ABCD is a parallelogram.
[Abbreviation: diags. bisect each other ] A B C D
Condition IV
One pair of opposite sides are equal
and parallel.
i.e. if AD = BC
and AD // BC,
then ABCD is a parallelogram.
[Abbreviation: opp. sides equal and // ]

13. In the figure, ABCD is a quadrilateral.
AB = DC = 3 cm
∵ B + C = 75 + 105
= 180
∴ AB // DC
∵ AB = DC and AB // DC
∴ ABCD is a parallelogram. A B C D
75
105
3 cm
given
int. s supp.
opp. sides equal and //

14. Follow-up question The figure shows a quadrilateral ABCD. (a) Find a.
(b) Prove that ABCD is a parallelogram.
55
2a + 15
70 + a a B C
Solution
(a) ∵ Sum of the interior angles of a quadrilateral is 360.
∴ (2a + 15) + a + (70 + a) + 55 = 360
4a + 140 = 360
4a = 220
a = 55

15. Follow-up question (cont’d)A D
The figure shows a quadrilateral ABCD.
(a) Find a.
(b) Prove that ABCD is a parallelogram.
55
2a + 15
70 + a a B C
Solution
A = 2a + 15 = 2(55) + 15 = 125
B = a = 55
C = 70 + a = 70 + 55 = 125
D = 55
∴ A = C and B = D
∴ ABCD is a parallelogram.
given
opp. s equal