PROPERTIES OF PARALLELOGRAM
About "Properties of parallelogram"
Properties of parallelogram :
The following are the three important properties of triangle.
1. In a parallelogram, the opposite sides are equal.
2. In a parallelogram,the opposite angles are equal.
3. The diagonals of parallelogram bisect each other.
Some other properties of triangle
- The diagonal of a parallelogram divides it into two triangles of equal area.
- A parallelogram is a rhombus if its diagonals are perpendicular
- Parallelograms are on same base and between same parallels are equal in area.
Properties of parallelogram - Practice problems
Problem 1 :
In the parallelogram ABCD if ∠A = 65°, find ∠B, ∠C and ∠D
Solution :
Let ABCD be a parallelogram in which ∠A = 65°
Since AD ∥ BC we can treat AB as a transversal, so
∠A + ∠B = 180°
65° + ∠B = 180°
∠B = 180° - 65°
∠B = 115°
According to the properties of parallelogram, the opposite angles are equal. So,
∠A = ∠C
∠B = ∠D
Therefore ∠B, ∠C and ∠D are 115°, 65° and 115° respectively.
Let us look at the next problem on "Properties of parallelogram"
Problem 2 :
In the parallelogram ABCD if ∠A = 108°, find ∠B, ∠C and ∠D
Solution :
Since AD ∥ BC we can treat AB as a transversal, so
∠A + ∠B = 180°
108° + ∠B = 180°
∠B = 180° - 108°
∠B = 72°
According to the properties of parallelogram, the opposite angles are equal. So,
∠A = ∠C
∠B = ∠D
Therefore ∠B, ∠C and ∠D are 72°, 108° and 72° respectively.
Let us look at the next problem on "Properties of parallelogram"
Problem 3 :
ABCD is a parallelogram ∠BAO = 30°, ∠DAO = 45° and ∠COD = 105°. Calculate
(i) ∠ ABO (ii) ∠ ODC (iii)∠ ACB (ii) ∠ CBD
Solution :
(i) ∠ DOC = ∠ AOB (vertically opposite angles are equal)
∠ AOB = 105°
In triangle AOB,
∠ AOB + ∠ OAB + ∠ ABO = 180°
105° + 30° + ∠ ABO = 180°
∠ ABO = 180° - 135°
∠ ABO = 45°
(ii) Since AB ∥ DC we can treat BD as a transversal, so
∠ ABO = 45° then
∠ ODC = 45° (alternate angles are equal)
(iii) Since AD ∥ BC we can treat AC as a transversal, so
∠ DAC = 45° then
∠ ACB = 45° (alternate angles are equal)
(iv) In triangle DCB,
∠ BDC + ∠ DCB + ∠ CBD = 180°
45° + 75° + ∠ CBD = 180°
∠ CBD = 180° - 120°
∠ CBD = 60°
Problem 4 :
Find the measure of each angle of a parallelogram in which AB = 9 cm and it perimeter is 30 cm. Find the length of each side of a parallelogram.
Solution :
Perimeter of parallelogram = AB + BC + CD + DA (length of all sides)
Perimeter = 30
Since it is parallelogram,length of opposite sides are equal.
9 + BC + 9 + BC = 30
2 BC = 30 - 18
2 BC = 12
BC = 6 cm.
Therefore length of AB and CD = 9 cm and length of BC and DA = 6 cm.
Problem 5 :
In the given figure find the values of x , ∠A and ∠C.
Solution :
According to the properties of parallelogram, opposite angles are equal.So,
∠D = ∠B
87° = (x + 29)°
87 - 29 = x
x = 58°
∠D + ∠C = 180°
(Adjacent angles are supplementary)
87° + ∠C = 180°
∠C = 180° - 87° = 93°
∠A = 93°
Therefore x = 58° , ∠A = 93° , ∠C = 93°
Problem 6 :
In the given figure AO = x + 40 and OC = 2x + 18. Find the length of AO and OC.
Solution :
AO = x + 40 and OC = 2x + 18
According to the properties of parallelogram, the diagonals bisect each other.So,
AO = OC
x + 40 = 2 x + 18
2x - x = 40 - 18
x = 22
AO = 22 + 40 = 62
OC = 2(22) + 18 = 44 + 18 = 62
Therefore AO and CO are 62 cm
Problem 7 :
In a parallelogram, one angle is four times greater than the other. Find the angles of the parallelogram.
Solution :
Let "x" be one angle,
then "4x" be the other angle
x + 4 x = 180
5 x = 180 ==> x = 36
then 4 x = 4 (36) = 144
Therefore the angles are 36°, 144°, 36° and 144°
Problem 8 :
In the given parallelogram, find the measures of the sides GJ and HI (in cm).
Solution :
According to the properties of parallelogram, the length of opposite sides are equal.
Length of CJ = Length of HI
x + 44 = 55
x = 55 - 44
x = 11
Length of CJ and HI = 11 + 44 = 55
Therefore CJ and HI are 55 cm
Problem 9 :
In the given parallelogram, find the measures of x and y
Solution :
According to the properties of parallelogram, the digonals of a parallelogram bisect each other.
x + y = 2y - 2 --------->(1)
3x = 2y --------->(2).
from (1)
x = 2y - 2 - y ===> x = y - 2
plug x = y - 2 in the second equation
3(y - 2) = 2 y ===> 3y - 6 = 2y ===> 3y - 2y = 6 ===> y = 6
plug y = 6 in x = y - 2
x = 6 - 2 ===> x = 4
Therefore x = 4 and y = 6
Problem 10 :
In the given parallelogram, find the measures of x and y
Solution :
∠EFY + ∠EYD + ∠EYF = 180°
7 x - 5 + 45 + 70 = 180°
7x + 110 = 180°
7 x = 180 - 110 ==> 7 x = 70 ==> x = 10
In a parallelogram opposite angles are equal
∠EYD = ∠EDY
7 x - 5 = 5 y
7(10) - 5 = 5 y
70 - 5 = 5 y ==> 65 = 5 y ==> y = 13
Therefore x = 10 and y = 13
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