PROPERTIES OF PARALLELOGRAM
The following are the four important properties of parallelogram.
1. The diagonals of a parallelogram bisect each other.
2. In a parallelogram, the opposite angles are congruent.
3. In a parallelogram, the opposite sides are equal.
4. In a parallelogram, the opposite sides are parallel.
Some Other Properties of Parallelogram
- 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 parallel lines are equal in area.
Practice Problems
Problem 1 :
In the parallelogram given below, find ∠B, ∠C and ∠D.
Solution :
In a parallelogram, adjacent angles are supplementary.
In the above parallelogram, ∠A and ∠B are adjacent angles.
So, we have
∠A + ∠B = 180°
65° + ∠B = 180°
∠B = 180° - 65°
∠B = 115°
Because opposite angles are congruent, we have
|
∠C = ∠A ∠C = 65° |
∠D = ∠B ∠D = 115° |
Hence, the measures of ∠B, ∠C and ∠D are 115°, 65° and 115° respectively.
Problem 2 :
In the parallelogram ABCD given below, find ∠A, ∠B, ∠C and ∠D.
Solution :
In a parallelogram, adjacent angles are supplementary.
In the above parallelogram, ∠A and ∠B are adjacent angles.
So, we have
x + 2x = 180°
3x = 180°
x = 60°
The measure of angle ∠A is
= x
= 60°
The measure of angle ∠B is
= 2x
= 2 ⋅ 60°
= 120°
According to the properties of parallelogram, the opposite angles are congruent.
So, we have
|
∠C = ∠A ∠C = 60° |
∠D = ∠B ∠D = 120° |
Hence, the measures of ∠A, ∠B, ∠C and ∠D are 60°, 120°, 60° and 120° respectively.
Problem 3 :
In the parallelogram given below, find the measures of ∠ABO and ∠ACB.
Solution :
In the parallelogram given above ∠AOB and ∠COD are vertically opposite angles.
Because vertically opposite angles are equal, we have
∠AOB = ∠COD
∠AOB = 105°
In triangle ABO, we have
∠OAB + ∠AOB + ∠ABO = 180°
Plug ∠OAB = 30° and ∠AOB = 105°.
30° + 105° + ∠ABO = 180°
135° + ∠ABO = 180°
∠ABO = 45°
In the parallelogram given above, AD||BC, AC is transversal and ∠OCB and ∠OAD are alternate interior angles.
If two parallel lines are cut by a transversal, alternate interior angles are equal.
So, we have
∠OCB = ∠OAD
In the parallelogram given above, ∠OAD = 45°.
So, we have
∠OCB = 45°
Because ∠OCB ≅ ∠ACB, we have
∠ACB = 45°
Hence, the measures of ∠ABO and ∠ACB are 45° each.
Problem 4 :
The perimeter of the parallelogram ABCD is 30 units and the length of the side AB is 9 units, find the length of other sides of the parallelogram.
Solution :
Given : Perimeter of the parallelogram is 30 units.
That is,
AB + BC + CD + AD = 30 -----> (1)
Because it is parallelogram, length of opposite sides must be equal.
So, we have
AB = CD
AD = BC
Because AB = 9 units and AB = CD, we can have
Then, we have
(1)-----> 9 + BC + 9 + AD = 30
18 + BC + AD = 30
Subtract 18 from both sides.
BC + AD = 12
Because AD = BC, we can have
AD + AD = 12
2 ⋅ AD = 12
Divide both sides by 2.
AD = 6
Then, the length of CD is also 6 units.
Hence, the length of CD is 9 units, AD and BC are 6 units each.
Problem 5 :
In the parallelogram given below, find the measures of ∠A and ∠C.
Solution :
According to the properties of parallelogram, opposite angles are equal.
So, we have
∠B = ∠D
(x + 29)° = 87°
x + 29 = 87
x = 58
In a parallelogram, adjacent angles are supplementary.
So, we have
∠D + ∠C = 180°
87° + ∠C = 180°
∠C = 93°
In a parallelogram, opposite angles are equal
So, we have
∠A = ∠C
∠A = 93°
Hence, the measures of ∠A and ∠C are 93° each.
Problem 6 :
In the parallelogram given below,
AO = x + 40
OC = 2x + 18
Find the length of AO and OC.
Solution :
According to the properties of parallelogram, the diagonals bisect each other.
So, we have
AO = OC
x + 40 = 2 x + 18
2x - x = 40 - 18
x = 22
The length of AO is
AO = x + 40
AO = 22 + 40
AO = 62
The length of OC is
OC = 2x + 18
OC = 2⋅ 22 + 18
OC = 44 + 18
OC = 62
Hence, the lengths of AO and OC is 62 units each.
Problem 7 :
In two adjacent angles of a parallelogram, if one angle is four times of the other, then find the measures of the two angles.
Solution :
Let "x" be one of the angles.
Then, the adjacent angle of x is 4x.
In a parallelogram, adjacent angles are supplementary.
So, we have
x + 4 x = 180°
5x = 180°
Divide both sides by 5.
x = 36°
Then, the measure of the adjacent angle is
= 4x
= 4 ⋅ 36°
= 144°
Hence, the measures of the two adjacent angles are 36° and 144°.
Problem 8 :
In the parallelogram given above, find the lengths of the sides GJ and HI (in cm).
Solution :
According to the properties of parallelogram, the length of opposite sides are equal.
Length of GJ = Length of HI
x + 44 = 5x
44 = 4x
11 = x
The length of HI is
= 5x
= 5 ⋅ 11
= 55
Because opposite sides are equal, the length of GJ is also 55 units.
Hence, the lengths of GJ and HI is 55 units each.
Problem 9 :
In the given parallelogram, find the values of x and y.
Solution :
According to the properties of parallelogram, the diagonals of a parallelogram bisect each other.
From the one of the diagonals, we have
x + y = 2y - 2
x = y - 2 -----> (1)
From the other diagonal, we have
3x = 2y -----> (2)
Plug x = y - 2 in (2).
(2)-----> 3(y - 2) = 2 y
3y - 6 = 2y
y = 6
Plug y = 6 in (1).
(1)-----> x = 6 - 2
x = 4
Hence, the values of x is 4 and y is 6.
Problem 10 :
In the parallelogram given below, find the values of x and y.
Solution :
In the parallelogram given above, the measure of angle Y is
∠Y = 45° + 70°
∠Y = 115°
In a parallelogram, adjacent angles are supplementary.
Because ∠F and ∠Y are supplementary, we have
∠F + ∠Y = 180°
Plug ∠F = 7x - 5 and ∠Y = 115°
7x - 5 + 115 = 180
7x + 110 = 180
7x = 70
x = 10
The measure of angle ∠F is
= (7x - 5)°
= (7 ⋅ 10 - 5)°
= (70 - 5)°
= 65°
In a parallelogram, opposite angles are equal.
So, we have
∠D = ∠F
(5y)° = 65°
5y = 65
y = 13
Hence, the value of x is 10 and y is 13.
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