Problem 1 :
In the parallelogram given below,find∠B,∠C and∠D.
Problem 2 :
In the parallelogram ABCD given below, find∠A,∠B,∠C and∠D.
Problem 3 :
In the parallelogram given below, find the measures of∠ABO and∠ACB.
Problem 4 :
The perimeter of the parallelogram ABCD shown below is 30 units and the length of the side AB is 9 units, find the length of other sides of the parallelogram.
Problem 5 :
In the parallelogram given below, find the value of x, measures of∠A and∠C.
Problem 6 :
In the parallelogram given below,
AO = x + 40
OC = 2x + 18
Find the length of AO and OC.
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.
Problem 8 :
In the parallelogram given above, find the lengths of the sides AB and CD.
Problem 9 :
In the parallelogram given below, find the values of x and y.
Problem 10 :
In the parallelogram given below, find the values of x and y.
1. Answer :
In a parallelogram, adjacent angles aresupplementary.
In the above parallelogram,∠A and∠B are adjacent angles.
∠A +∠B = 180°
65° +∠B = 180°
∠B = 115°
Becauseopposite angles are congruent, we have
∠C = ∠A ∠C = 65° | ∠D = ∠B ∠D = 115° |
Hence, the measures of∠B,∠C and∠D are115°,65° and115° respectively.
2. Answer :
In a parallelogram, adjacent angles aresupplementary.
In the above parallelogram,∠A and∠B are adjacent angles.
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.
∠C = ∠A ∠C = 60° | ∠D = ∠B ∠D = 120° |
Hence, the measures of∠A,∠B,∠Cand∠D are 60°, 120°, 60°and 120° respectively.
3. Answer :
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°
Substitute∠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.
∠OCB =∠OAD
In the parallelogramgiven above,∠OAD = 45°.
Then,
∠OCB =45°
Because∠OCB≅∠ACB, we have
∠ACB = 45°
Hence, the measures of∠ABO and∠ACB are45° each.
4. Answer :
Given : Perimeter of the parallelogram is 30 units.
AB + BC + CD + AD = 30 ----(1)
Because it is parallelogram, length of opposite sides must be equal.
AB = CD
AD = BC
Given : AB = 9 units.
In a parallelogram, opposite sides are equal, so AB = CD.
AB = CD = 6.
(1)----> 9 + BC + 9 + AD = 30
18 + BC + AD = 30
BC + AD = 12
Because AD = BC,
AD + AD = 12
2⋅ AD = 12
AD = 6
Then, the length of BC is also 6 units.
Hence, the length of CD is 9 units, AD and BC are 6 units each.
5. Answer :
According to the properties of parallelogram, opposite angles are equal.
∠B = ∠D
(x + 29)° = 75°
x + 29 =75
x = 46
In a parallelogram, adjacent angles are supplementary.
∠D +∠C = 180°
75° +∠C = 180°
∠C = 105°
In a parallelogram, oppositeangles are equal.
∠A =∠C
∠A = 105°
Hence, the measures of∠A and∠C are 105° each.
6. Answer :
AO = x + 40
OC = 2x + 18
According to the properties of parallelogram, the diagonals bisect each other.
AO = OC
x + 40 = 2x + 18
40 = x + 18
x = 22
Length of AO :
AO = x + 40
AO = 22 + 40
AO = 62
Length of OC :
OC = 2x + 18
OC = 2⋅22 + 18
OC = 44 + 18
OC = 62
Hence, the lengths of AO and OC are 62 units each.
7. Answer :
Let x be one of the angles.
Then, the adjacent angle of x is 4x
In a parallelogram, adjacent angles are supplementary.
x + 4x = 180°
5x = 180°
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°.
8. Answer :
According to the properties of parallelogram, the length of opposite sides are equal.
Length of AB = Length of CD
5x = x + 44
4x = 44
x = 11
Length of AB:
AB = 5x
= 5⋅ 11
= 55
Because opposite sides are equal, the length of CD is also 55 units.
Hence, the lengths of AB and CD are 55 units each.
9.Answer :
According to the properties of parallelogram, the diagonals of a parallelogram bisect each other.
From the diagonal AC, we have
x + y = 2y - 2
x = y - 2 ----(1)
From the diagonal BD, we have
3x = 2y ----(2)
Substitute x = y - 2 in (2).
3(y - 2) = 2y
3y - 6 = 2y
y = 6
Substitute y = 6 in (1).
x = 6 - 2
x = 4
Hence, the value of x is 4 and y is 6.
10. Answer :
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°
Substitute∠F = (7x - 5)°and∠Y = 115°.
(7x - 5)° + 115° = 180°
7x - 5 + 115 = 180
7x + 110 = 180
7x = 70
x = 10
The measure of angle∠F :
= (7x - 5)°
= (7⋅ 10 - 5)°
= (70 - 5)°
= 65°
In a parallelogram, opposite angles are equal.
∠D = ∠F
(5y)° =65°
5y = 65
y = 13
Hence, the value of x is 10 and y is 13.
Kindly mail your feedback tov4formath@gmail.com
We always appreciate your feedback.
©All rights reserved. onlinemath4all.com
