Understanding Quadrilaterals Test

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Understanding Quadrilaterals Test - 7

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Question 1
1 / -0

PQRS is a rhombus in which side PS is bisected by altitude from point R which intersects PS at M. Then, the angles P and S respectively are

A
120° and 60°

B
50° and 130°

C
60° and 120°

D
130° and 50°

Solution

In the given rhombus PQRS, RM ⊥ PS such that PM = MS.
Join P and R.
Let PM = MS = a
PS = RS = 2a

In triangle RMS,
RM² = RS² - MS²
= (2a)² - a²= 4a²- a²= 3a²
In triangle PMR,
PR²= RM² + PM²
= 3a² + a² = 4a²PR = 2a

and PR = RS = PS = 2a

Therefore, PQR is an equilateral triangle.

So, ∠PSR = 60°
Now, ∠PSR + ∠SPQ = 180°
∠SPQ = 180° - 60°
= 120°
So, ∠P = 120°
∠S = 60°
Question 2
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If the interior angle of a regular polygon is one half of its exterior angle, then find the number of sides.

A
4

B
2

C
3

D
8

Solution

If n be the number of sides, then
Exterior angle =

Interior angle = ( × 90^o)

Now, according to the question

= ( × 90^o)

2n - 4 = 2
2n = 6
n = 3
Question 3
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If a quadrilateral has four congruent sides and four right angles, then it is a

A
rectangle

B
rhombus

C
kite

D
square

Solution

If a quadrilateral has four congruent sides and four right angles, then it is a square.
Question 4
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If number of sides in a regular polygon are 9, then find the interior angle.

A
90^o

B
140^o

C
120^o

D
100^o

Solution

× 90° = Interior angle (where n is the number of sides)

Since n = 9
So, interior angle =
= 140^o
Question 5
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The measures of two adjacent angles of a quadrilateral are 130° and 30° and its other two angles are equal. Find the measure of each of the equal angles.

A
110°

B
100°

C
80°

D
120°

Solution

Let ABCD be the quadrilateral such that ∠A = 130°, ∠B = 30° and .

By angle sum property of a quadrilateral, we have
= 360^o

130^o + 30^o + = 360^o []

160^o + 2 = 360^o
= 360^o - 160^o = 200^o
= 100^o

Hence, = 100^o
Question 6
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In which of the following quadrilaterals only two pair of adjacent sides are equal and diagonals bisect each other at 90°?

A
square

B
parallelogram

C
kite

D
rhombus

Solution

In a kite, only two pair of adjacent sides are equal and diagonals bisect each other at 90°.
Hence, option (3) is correct.
Question 7
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What is the name of the given figure, if the non-parallel sides are equal?

A
Square

B
Kite

C
Isosceles trapezium

D
Trapezium

Solution

If a trapezium has non-parallel sides, then it is an isosceles trapezium.
Question 8
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If 1 side of a square is 6x and its perimeter is 240 cm, then find the length of sides.

A
120 cm

B
60 cm

C
40 cm

D
50 cm

Solution

In a square, all sides are equal.
According to the question, one side is 6x.
Therefore,
6x + 6x + 6x + 6x = 240
24x = 240
So, x = 10
Now, as all the sides are 6x, that means 6 × 10 = 60 cm is the length of all the sides of square.
Question 9
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ABCD is a square. A straight line CPQ cuts BD at P and BA is produced at Q. If CPD = 80°, then CQA is equal to

A
35°

B
55°

C
45°

D
10°

Solution

BPQ = 80°[Vertically opposite angles]
ABD = 45°[Property of a square]
Using angle sum property in BPQ,
BQP = 55°
Now, CQA = BQP = 55°
Question 10
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Which of the following is a convex polygon?

A

B

C

D

Solution

A simple polygon is strictly convex, if every internal angle is less than 180°.
In option (3), each angle is less than 180°.
Question 11
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Which of the following statements is/are true?

(i) A parallelogram with sides of equal length is called a rhombus.
(ii) If the diagonals of a quadrilateral are of equal length, then it is a rectangle.

A
Both (i) and (ii)

B
Only (i)

C
Only (ii)

D
Neither (i) nor (ii)

Solution

(i) If all the sides of a parallelogram are equal in length, then it is called a rhombus.
(ii) If the diagonals of a quadrilateral are of equal length, then it is a rectangle or a square. A square is also a special rectangle.
Hence, both (i) and (ii) are true.
Hence, (1) is the correct option.
Question 12
1 / -0

One angle of a quadrilateral is 120 degrees. The remaining angles are equal. Find the measure of each of the three remaining equal angles.

A
80 degrees

B
82 degrees

C
120 degrees

D
78 degrees

Solution

Let ABCD be the quadrilateral such that = 120^o and .

Also, let = x^oNow, by angle sum property of a quadrilateral, we have

= 360^o

Or, 120^o + x + x + x = 360^o

Or, 120^o + 3x = 360^o

Or, 3x = 240^o

Or, x = 80^o

Hence, the measure of each of the three remaining equal angles is 80^o.
Question 13
1 / -0

A parallelogram whose diagonals does not bisect each other at 90° but each angle measures 90° is called

A
square

B
rhombus

C
rectangle

D
kite

Solution

A parallelogram whose diagonals does not bisect each other at 90° but each angle measures 90° is called a rectangle.
Question 14
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If n is the number of sides of polygon, then the formula for calculating sum of interior angle in any polygon is

A
(n + 2) × 180°

B
n × 180°

C
(n - 2) × 180°

D
(n - 2) × 360°

Solution

If n is the number of sides of polygon, then the formula for calculating sum of interior angle in any polygon is (n - 2) × 180°.
Question 15
1 / -0

The quadrilateral having only one pair of opposite sides parallel is called

A
rhombus

B
trapezium

C
parallelogram

D
kite

Solution

The quadrilateral having only one pair of opposite sides parallel is called trapezium.
Question 16
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A garden is built in the shape of a kite and its perimeter is 210 m. If one of the sides is 45 m, what could be the possible lengths of the other sides?

A
45 m, 60 m, 60 m

B
45 m, 50 m, 50 m

C
45 m, 40 m, 40 m

D
45 m, 55 m, 55 m

Solution

In a kite, the two adjacent sides are of equal length.

So, we have x + x + y + y = 210
2x + 2y = 210
x + y = 105
Let the given side of 45 m be 'x'.
45 + y = 105
y = 60 m
So, the other sides are 45 m, 60 m and 60 m.
Question 17
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Hitesh's toy is in the shape of a rectangle whose one side is 6.4 cm and other side istimes of this side. He wants to wrap his toy with a gift paper four times. Find the length of the gift paper required.

A
114.1 cm

B
115.2 cm

C
116.3 cm

D
117.4 cm

Solution

Let ABCD be the rectangular toy.

We know that opposite sides of the rectangle are equal.
AC = BD = 6.4 cm
AB = CD =× 6.4
= × 6.4 = 8 cm
Perimeter = (8 + 6.4 + 8 + 6.4) cm = 28.8 cm
He wants to wrap gift paper around the toy four times.
So, length of the gift paper required
= 4 × 28.8 cm
= 115.2 cm
Question 18
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A park is in the form of a parallelogram whose perimeter is 320 m. One of its sides is 55 m. What is the difference between the lengths two unequal sides?

A
35 m

B
40 m

C
50 m

D
55 m

Solution

Let ABCD be the park which is in the form of a parallelogram.

Let AD = 55 m
Opposite sides of parallelogram are of equal length.
AD = BC = 55 m
Perimeter of parallelogram
ABCD = AB + BC + CD + DA
320 = (AB + CD) + 55 + 55
AB + CD = (320 - 110) = 210 m
2AB = 210
AB = 105 m = CD
Hence, difference between the lengths of two unequal sides = 105 - 55 = 50 m
Question 19
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Sajjan is trying to draw a regular polygon of 10 sides. He has to mark the lines at what internal angle, so that each line joins with the other line?

A
111°

B
122°

C
133°

D
144°

Solution

Each interior angle = , where n = Number of sides of the polygon
Here, n = 10
Each interior angle = = 144°
Question 20
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In a square PQRS, the diagonals bisect each other at O. Identify the type of triangle POQ.

A
An equilateral triangle

B
An isosceles but not right angled triangle

C
A right-angled but not an isosceles triangle

D
An isosceles right-angled triangle

Solution

In a square, diagonals bisect each other at 90^o.
POQ is an isosceles right-angled .
∠OPQ = ∠OQP = 45^o∠POQ = 90^o and OP = OQ

Question 21

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Directions: Match the following.

Column - 1	Column - 2
A. Parallelogram	1. One pair of opposite sides is parallel
B. Trapezium	2. There are two pairs of parallel sides
C. Rectangle	3. Diagonals bisect each other at 90°
D. Kite	4. All the angles are of the same measure

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. In a parallelogram, there are two pairs of parallel sides.
B. In a trapezium, one pair of opposite sides is parallel.
C. In a rectangle, all the angles are of the same measure.
D. In a kite, diagonals bisect each other at 90^o.
Hence, option (4) is correct.

Question 22
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Find the length of the side of a rhombus, if the lengths of its diagonals are 16 cm and 12 cm.

A
10 cm

B
8 cm

C
20 cm

D
12 cm

Solution

ABCD is a rhombus.

AC = 16 cm
AO = OC = 8 cm
and BD = 12 cm
BO = OD = 6 cm
Using Pythagoras Theorem, we get
AO² + OB² = AB²
64 + 36 = AB²Therefore, AB = 10 cm

Question 23

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Directions: Fill in the blanks.

(I) A quadrilateral having two pairs of equal and parallel sides is called D .
(II) In a C polygon, all the sides are equal.
(III) The sum of exterior angles of a quadrilateral is always B .
(IV) The sum of measure of interior angles of a polygon with N sides is A .

	A	B	C	D
P	(N - 1) × 180^o	360^o	Regular	Parallelogram
Q	(2N - 1) × 90^o	180^o	Irregular	Rhombus
R	(N - 2) × 180^o	360^o	Regular	Parallelogram
S	N × 180^o	360^o	Irregular	Rhombus

R

S

P

Q

Solution

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

Question 24
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ABCD is a trapezium with ∠ABD = 65°, ∠ABC = 135° and ∠ADC = 115°. If BD || CE, find the measures of ∠ECF and ∠DCE, respectively.

A
70°, 45°

B
70°, 30°

C
80°,45°

D
65°, 30°

Solution

Given: ∠ABD = 65°and ∠ABC = 135°
∠DBC = 135 – (65) = 70°Since DB || CE;
∠DBC = ∠ECF = 70°So, ∠ECF = 70°
Now, since AD || BC;
∠ADB = ∠DBC = 70°
Also, ∠ADC = 115°= ∠ADB + ∠BDC
So, ∠BDC = 115° - 70° = 45°
Now, BD || CE
∠BDC = ∠DCE = 45°(Alternate interior angles)
Question 25
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Directions: Select the incorrect statement.

A. If the diagonals of parallelogram are equal, then it is a rhombus.
B. The sum of interior angles in any quadrilateral is 360^o.
C. In a parallelogram, diagonals bisect each other at right angles.
D. The base angles of an isosceles trapezium are equal.

A
A and C

B
B and C

C
A and B

D
C and D

Solution

A. If the diagonals of parallelogram are equal, then it is a rhombus. This statement is incorrect because if the diagonals of parallelogram are equal, then it is a rectangle.
B. The sum of angles in any quadrilateral is 360^o. This statement is true.
C. In a parallelogram, diagonals bisect each other at right angles. This statement is incorrect. As in a parallelogram, diagonals does not bisect each other at right angles.
D. The base angles of an isosceles trapezium are equal. This statement is true.

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Understanding Quadrilaterals Test - 7

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Understanding Quadrilaterals Test - 7

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SHARING IS CARING

Question 1

120° and 60°

50° and 130°

60° and 120°

130° and 50°

Solution

Question 2

4

2

3

8

Solution

Question 3

rectangle

rhombus

kite

square

Solution

Question 4

90o

140o

120o

100o

Solution

Question 5

110°

100°

80°

120°

Solution

Question 6

square

parallelogram

kite

rhombus

Solution

Question 7

Square

Kite

Isosceles trapezium

Trapezium

Solution

Question 8

120 cm

60 cm

40 cm

50 cm

Solution

Question 9

35°

55°

45°

10°

Solution

Question 10

Solution

Question 11

Both (i) and (ii)

Only (i)

Only (ii)

Neither (i) nor (ii)

Solution

Question 12

80 degrees

82 degrees

120 degrees

78 degrees

Solution

Question 13

square

rhombus

rectangle

90^o

140^o

120^o

100^o

80°,45°