Understanding Quadrilaterals Test

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Understanding Quadrilaterals Test - 6

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Question 1
1 / -0

There is a rhombus PQRS in which the altitude from S to side PQ bisects PQ. Then, ∠QPS and ∠PQR respectively are

A
45° and 90°

B
40° and 100°

C
50° and 100°

D
60° and 120°

Solution

In the given figure, PQRS is a rhombus and ST ⊥ PQ such that PT = TQ. Join QS.
Let PQ = QR = RS = SP = 2a
So, PT = a = TQ

In △PTS,
TS² = PS² - PT² (Pythagoras theorem)
= (2a)² - a² = 4a² - a² = 3a²

In △STQ, we have QS² = TQ² + TS² = a² + 3a² = 4a²
QS = 2a
So, PQ = QS = SP = 2a
△PQS is an equilateral triangle.

∠QPS = 60° and ∠PQS = 60°
Similarly, △QSR is an equilateral triangle and ∠SQR = 60°.
Now, ∠PQR = ∠PQS + ∠SQR = 60°+ 60° = 120

So, ∠QPS = 60° and ∠PQR = 120°
Question 2
1 / -0

How many sides will a polygon have if it's exterior angle is one-fifth of it's interior angle?

A
8

B
10

C
12

D
14

Solution

Let n be the number of sides of the polygon.

Each exterior angle =

And each interior angle =

According to questions, we have

Exterior angle = (Interior angles)

1800 = (2n - 4) × 90

1800 + 360 = 180n

2160 = 180n

n = 12
Question 3
1 / -0

If one pair of opposite sides is parallel in a quadrilateral, then it is a

A
parallelogram

B
trapezium

C
rectangle

D
rhombus

Solution

If one pair ofopposite sides is parallel in a quadrilateral, then it is a trapezium.
Question 4
1 / -0

The number of sides of a regular polygon whose each exterior angle measures of 60° is

A
12

B
10

C
8

D
6

Solution

Exterior angle of a regular polygon = 60°

Let the number of sides be n.

Then 60° =

n = = 6

n = 6
Question 5
1 / -0

In the given figure, the value of x is _______

A
120°

B
130°

C
140°

D
150°

Solution

In the given figure RSP and QRS = 90°
Now in quadrilateral PQRS, we have
SPQ + PQR + QRS + RSP = 360°
40° + x + 90° + 90° = 360°
x + 220° = 360°
x = 140°
Question 6
1 / -0

If all the angles and opposite sides are equal in a quadrilateral and the diagonals are also equal in length and intersect in the middle, then it is a

A
parallelogram

B
trapezium

C
rhombus

D
rectangle

Solution

If all the angles and opposite sides are equal in a quadrilateral and the diagonals are also equal in length and intersect in middle, then it is a rectangle.
Question 7
1 / -0

The given quadrilateral MNOP is a

A
convex quadrilateral

B
rectangle

C
concave quadrilateral

D
rhombus

Solution

A convex quadrilateral is a four sided polygon that has interior angles that measure less than 180° each.

Hence, option (1) is correct.
Question 8
1 / -0

The perimeter of a parallelogram is 40 cm and the ratio of it's two sides is 2 : 3. Find the lengths of the sides of the parallelogram.

A
5 cm, 10 cm

B
6 cm, 10 cm

C
8 cm, 12 cm

D
9 cm, 12 cm

Solution

Let ABCD be the parallelogram in which CD = 2x and BC = 3x.
Since opposite sides of a parallelogram are equal,
So, CD = AB = 2x cm and BC = DA = 3x cm
Perimeter of the parallelogram = 40 cm

AB + BC + CD + DA = 40 cm
(2x + 3x + 2x + 3x) = 40 cm
10x = 40 cm
x = 4 cm
So, lengths of the sides AB = DC = 2 × 4 = 8 cm and BC = AD = 3 × 4 = 12 cm.
Question 9
1 / -0

In the given figure, ∠MNO = 50°, ∠NOP = 120°, ∠OPM = 50° and ∠OMN = 60°. If line QP is drawn parallel to MO, then ∠QPM = _______.

A
60°

B
80°

C
90°

D
85°

Solution

In △MNO, we have
∠MON = 180° - (60° + 50°)
= 180° - 110°
= 70°

Now, ∠NOP = 120°
∠NOP = ∠MON + ∠MOP
120 ° = 70° + ∠MOP
∠MOP = 50°

Now, MO || QP
So ∠MOP = ∠QPR = 50°
And since ∠OPM = 50^°

∠QPM = 180° - (∠QPR + ∠OPM)
= 180° - (50° + 50°)
= 80°
Question 10
1 / -0

Which of the following quadrilaterals is a rhombus?

A
Square

B
Rectangle

C
Kite

D
Parallelogram.

Solution

A rhombus is a quadrilateral with all sides equal in length. A square is a quadrilateral with all sides equal in length and all interior angles are right angles. Thus, a rhombus is not a square unless the angles are all right angles. However, a square is a rhombus since all four of its sides are of the same length.
Question 11
1 / -0

Which of the following statements is correct?

A
In a rectangle opposite sides are of unequal lengths.

B
Diagonally opposite angles are not equal in a rhombus.

C
One pair of opposite sides is parallel in a trapezium.

D
Only one pair of sides are of equal length in a kite.

Solution

It is true that one pair of opposite sides is parallel in a trapezium.
Question 12
1 / -0

What will be the sum of the smallest and greatest angles of the quadrilateral, if it's angles are in the ratio of 2 : 3 : 3 : 4?

A
150°

B
160°

C
170°

D
180°

Solution

Let the angles be 2x, 3x, 3x, 4x.
Sum of the angles of a quadrilateral = 360°
2x + 3x + 3x + 4x = 360°
12x = 360°
x = 30°
Required sum of angles = 2x + 4x = 6x = 6(30°) = 180°
Question 13
1 / -0

A rhombus which only has right angles is said to be a

A
square

B
rectangle

C
parallelogram

D
kite

Solution

A rhombus which only has right angles is said to be a square as it will have all the properties of a square.

Hence, option (1) is correct.
Question 14
1 / -0

Which of the following can be the measure of exterior angle of a regular polygon?

A
27°

B
60°

C
37°

D
16°

Solution

60° can be the measure of exterior angle of a regular polygon as it is a factor of 360°, however other given angles cannot be for the same reason.

Hence, option (2) is correct.
Question 15
1 / -0

A quadrilateral with two distinct pairs of equal adjacent sides and having only one pair of diagonally opposite angles is called a

A
parallelogram

B
rhombus

C
kite

D
trapezium

Solution

The quadrilateral with only one pair of diagonally opposite angles equal as well as two distinct pairs of equal adjacent sides is called a kite.
Question 16
1 / -0

A park is in the shape of a triangle of perimeter 120 m. What could be the possible lengths of all the three sides?

A
30 m, 50 m, 40 m

B
80 m, 20 m, 20 m

C
20 m, 30 m, 70 m

D
25 m, 65 m, 30 m

Solution

The park is in the shape of a triangle; and in a triangle, the sum of any two sides is more than the third side.
Since perimeter is the sum of all the sides;
Possible sum = 30 + 50 + 40 = 120 m
Hence, the only possible sides are 30 m, 50 m and 40 m.
Question 17
1 / -0

Ram's terrace is in the shape of a rectangle whose one side is 5.8 m and other side is times of the first side. He wants to put lights along the sides three times. Find the length of wire required to put the lights.

A
82.8 m

B
81.2 m

C
78.9 m

D
75.2 m

Solution

Let ABCD be the terrace of the building in the shape of rectangle.
Opposite sides of the rectangle are equal.
AC = BD = 5.8 m

AB = CD = × 5.8 m
Ram wants to put light around the terrace, the total perimeter = 2 x ( × 5.8 + 5.8)
He wants to put lights three times
Length of wire required =3 x 2 x ( × 5.8 + 5.8) = 81.2 m wire is required.
Question 18
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A swimming pool is in the form of an isosceles trapezium whose perimeter is 470 cm. One of its non-parallel side is 40 cm. What is the sum of parallel sides?

A
380 cm

B
385 cm

C
390 cm

D
395 cm

Solution

Let ABCD be the swimming pool which is in the form of a trapezium,
AB = 40 cm(Given)
Non-parallel sides of an isosceles trapezium are of equal length
Perimeter of trapezium ABCD = AB + BC + CD + DA
470 = (BC + DA) + 40 + 40
BC + DA = 470 - 80 = 390 cm
Hence, the sum of parallel sides = 390 cm
Question 19
1 / -0

Sahil is trying to draw a regular polygon of 8 sides. At what angle does he have to mark the lines so that each line joins with other line?

A
100°

B
125°

C
130°

D
135°

Solution

Each interior angle =

Where n = Number of sides of the polygon

Here, n = 8
Each interior angle =

= 135°
Question 20
1 / -0

Rohan is playing in a garden which is in the shape of a quadrilateral. He observes that only one pair of opposite sides is parallel. So, what is the shape of the garden?

A
Square

B
Rectangle

C
Trapezium

D
Parallelogram

Solution

A square is a flat shape with 4 equal sides and every angle is a right angle. Also both pair of opposite sides are parallel.
A rectangle is a four-sided flat shape where every angle is a right angle. Also both pair of opposite sides are parallel and of equal length.
A parallelogram has both pair of opposite sides parallel and equal in length. Also opposite angles are equal.
Only trapezium has only 1 pair of opposite sides parallel. Hence, it is a trapezium.

Question 21

1 / -0

Match the following:

Column I	Column II
A. The diagonals of a rectangle	1. Pair of opposite sides is parallel
B. Trapezium is a quadrilateral in which a	2. Are equal and intersect at right angles
C. The diagonals of a rhombus	3. Equal and parallel
D. In a parallelogram, two pairs of sides are	4. Are not equal

A - 2, B - 1, C - 3, D - 4

A - 3, B - 1, C - 2, D - 4

A - 1, B - 2, C - 4, D - 3

A - 2, B - 1, C - 4, D - 3

Solution

(A) The diagonals of a rectangle are equal and intersect at right angles.
(B) Trapezium is a quadrilateral in which a pair of opposite sides is parallel.
(C) The diagonals of a rhombus are not equal.
(D) In a parallelogram, two pairs of sides are equal and parallel.

Question 22
1 / -0

In the given figure, OB is the angle bisector of ∠COA and the measure of ∠COB is given. Find the value of x.

A
15

B
20

C
21

D
30

Solution

∠COA = 90°
We know that OB is the angle bisector.
So, ∠COB = 45°= 2x + 3
2x + 3 = 45°
2x = 42°
x = 21°

Hence, option (3) is correct.

Question 23

1 / -0

Fill in the blanks with the help of the table.

(I) The sum of interior angles of a polygon with n sides is ___P____.
(II) The sum of exterior angles of a quadrilateral is always ___Q____.
(III) In a ___R___ polygon, all the sides are equal.
(IV) A quadrilateral having two pairs of equal and parallel sides is a ___S____.

	P	Q	R	S
A	(n - 2)90	360	regular	parallelogram
B	(2n - 1) × 90	180	irregular	quadrilateral
C	(n - 2)180	360	regular	parallelogram
D	n × 180	360	irregular	rhombus

C

A

B

D

Solution

(I) The sum of interior angles of a polygon with n sides is (n - 2) x 180°.
(II) The sum of exterior angles of a quadrilateral is always 360°.
(III) In a regular polygon, all the sides are equal.
(IV) A quadrilateral having two pairs of equal and parallel sides is a parallelogram.

Question 24
1 / -0

The lengths of diagonals of a rhombus are 20 and 48 cm. Find the perimeter of the rhombus.

A
115 cm

B
112 cm

C
105 cm

D
104 cm

Solution

ABCD is a rhombus.
ΔAOB is a right angled triangle. So, as per Pythagoras theorem,
AB² = 10² + 24²
AB = 26 cm
Similarly we can find BC, CD and AD, which will be 26 cm each.
So, perimeter = 4 × 26 = 104 cm
Question 25
1 / -0

Select the incorrect statement.

(A) Every square is a rectangle.
(B) The sum of angles in any quadrilateral is 360 degrees.
(C) In a rhombus, diagonals do not intersect at right angles.
(D) If the diagonals of a parallelogram are equal, then it is a rectangle.

A
A

B
C

C
B

D
D

Solution

(A) Every square is a rectangle because the sides are equal or two pairs of sides are equal.
(B) The sum of angles in any quadrilateral is 360 degrees. It is true.
(C) In a rhombus, diagonals do not intersect at right angles. This is false because in a rhombus, diagonals make an angle of 90 degrees.
(D) If the diagonals of a parallelogram are equal, then it is a rectangle. It can be proved by SSS rule, hence true.

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Understanding Quadrilaterals Test - 6

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Understanding Quadrilaterals Test - 6

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Question 1

45° and 90°

40° and 100°

50° and 100°

60° and 120°

Solution

Question 2

8

10

12

14

Solution

Question 3

parallelogram

trapezium

rectangle

rhombus

Solution

Question 4

12

10

8

6

Solution

Question 5

120°

130°

140°

150°

Solution

Question 6

parallelogram

trapezium

rhombus

rectangle

Solution

Question 7

convex quadrilateral

rectangle

concave quadrilateral

rhombus

Solution

Question 8

5 cm, 10 cm

6 cm, 10 cm

8 cm, 12 cm

9 cm, 12 cm

Solution

Question 9

60°

80°

90°

85°

Solution

Question 10

Square

Rectangle

Kite

Parallelogram.

Solution

Question 11

In a rectangle opposite sides are of unequal lengths.

Diagonally opposite angles are not equal in a rhombus.

One pair of opposite sides is parallel in a trapezium.

Only one pair of sides are of equal length in a kite.

Solution

Question 12

150°

160°

170°

180°

Solution