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Questions on Parallelogram
1. PQRS is a parallelogram in which ∠P = 110°. Find the measure of each of the angles ∠Q, ∠R, and ∠S.
Solution:
In a parallelogram PQRS, if ∠P = 110°, then ∠R = 110°.
The Sum of adjacent angles is supplementary in a parallelogram.
∠P + ∠Q = 180º.
Substitute ∠P value in the above equation.
110º + ∠Q = 180º
∠Q = 180º – 110º
∠Q = 70º
As the Opposite angles are equal in a parallelogram,
∠Q = ∠S = 70º
The final Answer is ∠Q = ∠S = 70º, ∠R = 110°.
2. Two adjacent angles of a parallelogram are equal. What is the measure of each of these angles?
Solution:
We know that Two adjacent angles of a parallelogram are equal.
Let the angle be m.
Adjacent angles of a parallelogram = 180
m + m = 180
2m = 180
m = 180/2
m = 90.
The final answer is every angle is 90º.
3. The ratio of two sides of a parallelogram is 10 : 8. If its perimeter is 108 cm, find the lengths of its sides?
Solution:
Let the lengths of two sides of the parallelogram be 10a cm and 8a cm respectively.
Find the perimeter using given values.
Then, its perimeter = 2(10a + 8a) cm = 2 (18a) cm = 36a cm.
Therefore, 36a = 108 ⇔ a = 108/36 = 3.
Therefore, one side = (10 × 3) cm = 30 cm and other side = (8 × 3) cm = 24 cm.
4. Two adjacent angles of a parallelogram are (6x + 15)° and (2x – 5)°. What is the value of x?
Solution:
The sum of the adjacent angles of parallelogram = 180
6x + 15 + 2x – 5 = 180
8x + 10 = 180
8x = 180 – 10
8x = 170
x = 170/8
The final answer is x = 21.25.
5. The length and breadth of a rectangle are in the ratio 16 : 12. If the diagonal measures 100 cm. Find the perimeter of a rectangle?
Solution:
Let m be the common multiple.
Length = 16m
Breadth = 12m
According to Pythagoras theorem,
(16m)² + (12m)²=(100)²
256m²+ 144m² = 10,000
400m² = 10,000
m² = 10,000/400
m = 25
So, Length = 16m = 400 cm
Breadth = 12m = 300 cm
Perimeter = 2 (l × b)
= 2 (400 + 300)
= 1400 cm
So, perimeter of rectangle is 1400 cm.
6. A rhombus has diagonals of 32 cm and 24 cm. Find the length of each side?
Solution:
Given that One diagonal is 32 and another 24 then half of both is 16 and 12. Diagonal of a rhombus bisect at 90º.
By pythogaurus theorem
h² = (16)² + (12)²
h² = 256 + 144 = 400
h = √400 = 20
Side = 20.
7. The sum of two opposite angles of a parallelogram PQRS is 130º find the measure of each of its angles.
Solution:
Given that the sum of two opposite angles of a parallelogram PQRS is 130º.
Let the angle be a. So, because the sum of two opposite angles is 130º.
a + a = 130
2a = 130
a = 65.
So, Again assume that angle is a
65 + 65 + a + a = 360
2a = 360 – 130
2a = 230
a = 115
So, 1st angle is 65
2nd angle is 65
3rd angle is 115
4th angle is 115.
8. In the adjacent figure, PQRS is a parallelogram and line segments PT and RV bisect the angles P and R respectively. Show that PT ∥ RV.
Solution:
In triangles PST and RQV, we have PS = QR, ∠Q = ∠S, and ∠SPT = ∠QRV.
[Since, ∠P = ∠R = ¹/₂∠P = ¹/₂∠R, ie., ∠SPT = ∠QRV]
Therefore, ∆ PST ≅ ∆RQV. And therefore, RS – ST = PQ – QV.
So, RT = PV.
Therefore, PTQV is a parallelogram.
Hence, PT ∥ RV.
9. The sides of a rectangle are in the ratio 5 : 4 and its perimeter is 90 cm. Find its length and breadth.
Solution:
The sides of a rectangle are in the ratio 5 : 4
Let the ratio be m.
so, Length = 5m
Breadth = 4m
Write the formula of the Perimeter of a rectangle and substitute the values in it.
Perimeter of rectangle = 2 (l + b) = 2(5m + 4m) = 2(9m) = 18m
Now we are given that its perimeter is 90 cm
So, 2 (9m) = 18m
18m = 90
m = 90/18 = 5
So, Length = 5m = 5 . 5 = 25
Breadth = 4m = 4 . 5 = 20
Hence its length and breadth are 25 cm and 20 cm respectively.
10. The perimeter of a parallelogram is 120 cm. If one of the sides is longer than the other by 10 cm, find the length of each of its a side
Solution:
Let the first side is a
then the second side becomes (a + 10)
According to the question 2(a + a + 10) = 120
2(2a + 10) = 120
2a + 10 = 60
2a = 60 – 10
2a = 50
a = 25
First side = a = 25cm
second side = a + 10 = 35cm
11. Name each of the following parallelograms.
(i) Name the parallelogram its diagonals are equal and the adjacent sides are unequal.
(ii) The diagonals are equal and the adjacent sides are equal.
(iii) Name the parallelogram which consists of all the sides that are equal and one angle is 90°.
(iv) All the sides are equal and one angle is 60°.
(v) The diagonals are unequal and the adjacent sides are equal.
(vi) All the angles are equal and the adjacent sides are unequal.
Solution:
(i) rectangle
(ii) square
(iii) square
(iv) rhombus
(v) rhombus
(vi) rectangle
12. State the below statements are true or false.
(i) The diagonals of a parallelogram are equal.
(ii) Every rhombus is a kite.
(iii) Every rectangle is a square.
(iv) The diagonals of a rectangle are perpendicular to each other.
(v) Every square is a rhombus.
(vi) Every square is a parallelogram.
(vii) The diagonals of a rhombus are equal.
(viii) Every rectangle is a parallelogram.
(ix) Every parallelogram is a rectangle.
(x) Check: Every rhombus is a parallelogram.
Solution:
(i) False
(ii) False
(iii) False
(iv) False
(v) True
(vi) True
(vii) False
(viii) True
(ix) False
(x) True