Quadrilaterals Test

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Quadrilaterals Test - 29

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Weekly Quiz Competition

Question 1
1 / -0

Two angles of a quadrilateral are equal and the other two angles are $${ 70 }^{ o}$$ and $${ 80 }^{o }$$. Then the equal angles are:

A
$${ 210 }^{\circ }$$

B
$${ 105 }^{\circ }$$

C
$${ 15 }^{\circ }$$

D
$${ 150 }^{\circ }$$

Solution

Let the four angles be $$ \angle A , \angle B , \angle C$$ and $$\angle D$$ .

Given two angles are equal.
Then $$ \angle A =\angle B=x $$.
Also, $$\angle C = 70^{o} $$ and $$\angle D = 80^{o} $$.

We know, by angle sum property, the sum of angles of a quadrilateral is $$360^o$$.
$$ \angle A + \angle B + \angle C + \angle D = 360^{o} $$
$$x + x + 70^o + 80^o = 360^{o} $$
$$2x+ 150^o = 360^{o} $$
$$ \Rightarrow 2x = 360 ^o-150^o $$

$$ \Rightarrow 2x = 210^o$$
$$ \Rightarrow x = 105^o$$.

Hence, the equal angles are $$ = 105^o$$ each
Therefore, option $$B$$ is correct.
Question 2
1 / -0

In the given figure, $$\angle ADC = ?$$

A
$$30^o$$

B
$$60^o$$

C
$$70^o$$

D
$$80^o$$

Solution

$$\angle ADC = ? $$

Given , $$\angle AEB = 60^o $$
$$\therefore \angle AEC =180^o - 60^o \\= 120^o\: $$

It is known that sum of interior angles angles of a quadrilateral is $$360^o$$

$$\therefore \angle ADC+ \angle DCE + \angle CEA+ \angle EAD = 360^o$$

$$\angle ADC = 360 - 120^o - 90^o - 90^o \:$$ (property of quadrilateral)
$$\angle ADC = 360 - 300 \\= 60^o $$
Question 3
1 / -0

Opposite angles of a quadrilateral $$PQRS$$ are equal. Find $$RS$$, if $$PQ = 5$$ cm.

A
$$5$$ cm

B
$$10$$ cm

C
$$4$$ cm

D
$$6$$ cm

Solution

If $$\angle 1 = \angle 3$$
& $$\angle 1 + \angle 2 + \angle 3 + \angle 4 = 360^o$$
$$\therefore \angle 2 + \angle 4 = 180^o$$
& $$ \angle 2 = \angle 4$$
$$\therefore$$ $$PQRS$$ is parallelogram
So the opposite sides are equal and parallel.
$$PQ = RS = 5$$ cm
Question 4
1 / -0

A Rhombus is also a

A
Parallelogram

B
Trapezium

C
rectangle

D
None of these

Solution

A Rhombus has two pairs of parallel side, opposite angles are equal and all the sides are equal, which are the properties of a parallelogram.
Hence, a rhombus is also a parallelogram.

Whereas, a Rhombus is not a trapezium as in a trapezium four siders are not equal.

Also, a Rhombus is not a rectangle as all the four siders are not equal.

Therefore, A is the correct answer.
Question 5
1 / -0

All squares are

A
rectangles

B
rhombus

C
parallelograms

D
All of these

Solution

($$1$$) Opposite sides are equal and angles are right angles so sqaure is a rectangle.
($$2$$) Diagonals bisect each other so square is a rhombus
($$3$$) Opposite sides are parallel so square is a parallelogram.
Correct answer is option (D).
Question 6
1 / -0

Identify the Quadrilateral.

A
parallelogram

B
rectangle

C
square

D
trapezium

Solution

It is a trapezium as it has no equal sides, no right angles, opposite angles are not equal. It cannot be a parallelogram, rectangle or square.
Therefore, D is the correct answer.
Question 7
1 / -0

In parallelogram, both pairs of opposite sides are ....................

A
equal

B
parallel and equal

C
perpendicular

D
perpendicular and equal

Solution

In $$ABCD$$ is a parallelogram,

$$\Rightarrow$$ We know that $$AB\parallel CD$$

Here $$AC$$ is a transversal for the parallel lines $$AB$$ and $$CD$$

$$\Rightarrow$$ So, $$\angle BAC = \angle DCA$$ [Alternate interior angles are equal] $$... ( 1 )$$

$$\Rightarrow$$ Similarly, we know that $$BC\parallel AD$$

$$\Rightarrow \angle BCA = \angle DAC$$ $$... ( 2 )$$

In $$\triangle ABC$$ and $$\triangle ADC,$$

We have $$\angle BAC = \angle DCA$$ [From ( 1 )]

$$\Rightarrow$$ $$AC$$ is the common side

$$\Rightarrow \angle BCA = \angle DAC$$ [From ( 2 )]

$$\Rightarrow$$ So, $$\triangle ABC \cong \triangle CDA$$ [As per ASA Congruence rule]

$$\therefore AB = CD$$ and $$BC = AD$$ [Corresponding sides of congruent triangle are equal ]

Hence, both pairs of opposite sides in a parallelogram are parallel and equal.
Question 8
1 / -0

A four sided figure with all sides equal and opposite angles also equal.

A
kite

B
Rhombus

C
Parallelogram

D
Rectangle

Solution

Among all, rhombus has all sides equal with opposite angles equal.
Therefore, B is the correct answer.
Question 9
1 / -0

Which statement(s) is/are correct?
(1)The diagonals of a parallelogram are equal.
(2)The diagonals of a square are perpendicular to each other.
(3)If the diagonals of a quadrilateral intersect at right angles, it is not necessarily a rhombus.
(4)Every quadrilateral is either a trapezium or a parallelogram or a kite.

A
Only (2)

B
Only (3)

C
Both (2) and (3)

D
(1), (2) and (3)

Solution

$$\Rightarrow$$ In four given statement we have given that, the diagonals of a parallelogram are equal which is not correct because we know that in properties of parallelogram it has opposite sides parallel but it's diagonals are not equal.
$$\Rightarrow$$ Second statement says, the diagonals of a square are perpendicular to each other which is true because we know that square has all four sides equal, opposite sides parallel to each other and both diagonals are equal and perpendicular to each other.
$$\Rightarrow$$ Third statement says that, if the diagonals of a quadrilateral intersect at right angles, it is not necessarily a rhombus, which is correct because the diagonals of a square also bisect each other at right angles.
$$\Rightarrow$$ Fourth statement says that, every quadrilateral is either a trapezium or a parallelogram or a kite which is wrong because rectangle, rhombus and square are also types of quadrilateral.
$$\Rightarrow$$ So correct statements are statement ( 2) and (3) then correct option is option C.
Question 10
1 / -0

ABCD is a quadrilateral such that $$AB = BC$$, $$AD= \cfrac{1}{2} CD$$ and $$AD = \cfrac{1}{4} AB$$. If $$BC =12$$ cm, what is the measures of AD?

A
$$6$$ cm

B
$$4$$ cm

C
$$12$$ cm

D
$$3$$ cm

Solution

$$\Rightarrow$$ In quadrlateral ABCD,
$$\Rightarrow$$ AB = BC [Given] --- ( 1 )
$$\Rightarrow$$ BC = 12cm. [Given]
$$\Rightarrow$$ AB = 12cm. [From ( 1 )]
$$\Rightarrow$$ It is given that, $$AD=\dfrac{1}{4} AB$$
$$\Rightarrow$$ $$AD=\dfrac{1}{4} \times 12$$ [Substituting value of AB]
$$\therefore$$ $$AD=3cm.$$