Quadrilaterals Test

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Quadrilaterals Test - 18

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Question 1
1 / -0

Diagonals of a quadrilateral $$ABCD$$ bisect each other. If $$\angle A=45^{\circ},$$ then $$\angle B$$ equals

A
$$45^{\circ}$$

B
$$55^{\circ}$$

C
$$135^{\circ}$$

D
$$115^{\circ}$$

Solution

The diagonal bisect each other. By parallelogram theorem, if the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
Hence, ABCD is a parallelogram.
In a parallelogram, sum of adjacent angles is 180
$$\angle A + \angle B = 180$$
$$\angle B = 180 - 45$$
$$\angle B = 135^{\circ}$$
Question 2
1 / -0

In the figure, PQRS is a parallelogram. If $$\angle P=75^o$$, then $$\angle Q=$$

A
$$75^o$$

B
$$90^o$$

C
$$105^o$$

D
$$100^o$$

Solution

In a $$\parallel gm $$ sum of adjacent angles $$=180^{\circ} $$
$$\angle P+\angle Q=180^{\circ}\Rightarrow 75^{\circ}+\angle Q=180^{\circ} $$
$$\angle Q=105^{\circ} $$
Question 3
1 / -0

In a quadrilateral ABCD, $$\angle A + \angle C = 180^{\circ}$$ then $$\angle B + \angle D = $$

A
$$360^{\circ}$$

B
$$100^{\circ}$$

C
$$180^{\circ}$$

D
$$80^{\circ}$$

Solution

We have,
$$\angle A + \angle B + \angle C + \angle D = 360^{\circ}$$
$$(\angle A + \angle C) + (\angle B + \angle D) = 360^{\circ}$$
$$180^{\circ} + (\angle B + \angle D) = 360^{\circ}$$
$$\angle B + \angle D = 180^{\circ}$$
Question 4
1 / -0

Which of the following is not true?

A
A parallelogram having a pair of adjacent sides equal, is called a rhombus

B
The diagonals of a rhombus do not bisect each other

C
If the diagonals of a parallelogram bisect each other at right angles, it is a rhombus

D
A rectangle is a parallelogram

Solution

We know that, the diagonals of a rhombus are perpendicular bisectors of each other.
Thus, the statement given in option B is false.

Hence, option B is correct.
Question 5
1 / -0

Find the angle measure $$x$$ in the following figures

A
$$140^o$$

B
$$120^o$$

C
$$180^o$$

D
$$220^o$$
Question 6
1 / -0

If the diagonals of a quadrilateral are perpendicular bisectors of each other, it is always a________

A
Rectangle

B
Rhombus

C
pentagon

D
Parallelogram

Solution

The diagonals of a Rhombus and Square bisect each other perpendicularly, so the quadrilateral should be
a Rhombus or a square, But for a quadrilateral to be a square, each angel in it should be $$90^{0}$$ , which is not given,
Also square is not given in the options, so the correct option is Rhombus.
Question 7
1 / -0

The sum of all angles in a quadrilateral is equal to _____ right angles.

A
$$2$$

B
$$3$$

C
$$4$$

D
$$360$$

Solution

Consider a quadrilateral $$ABCD$$.
Then, $$\angle A+\angle B+\angle C+\angle D=360^{\circ}$$.
To prove this, we join $$A$$ and $$C$$, i.e. we draw the diagonal $$AC$$.
In $$\triangle ABC$$,
$$\angle CAB+\angle ABC+\angle BCA=180^{\circ}$$ [Sum of all angles of a triangle is $$180^{\circ}$$].....$$(1)$$.
In $$\triangle ACD$$,
$$\angle CAD+\angle ADC+\angle DCA=180^{\circ}$$ [Sum of all angle of a triangle is $$180^{\circ}$$]....$$(2)$$.
Adding $$(1)$$ and $$(2)$$, we get,
$$\left(\angle CAB+\angle ABC+\angle BCA \right)+ \left(\angle CAD+\angle ADC+\angle DCA \right)=180^{\circ}+180^{\circ}$$
$$\implies$$ $$\angle ABC+\angle ADC+(CAB+CAD)+(BCA+DCA)=360^{\circ}$$
$$\implies$$ $$\angle ABC+\angle ADC+\angle BAD+\angle BCD=360^{\circ}$$
$$\implies$$ $$\therefore \angle A+\angle B+\angle C+\angle D=360^{\circ}$$.
That is, the sum of all angles of a quadrilateral is $$360^o=4\times90^o$$, i.e. $$4$$ right angles.
Therefore, option $$C$$ is correct.
Question 8
1 / -0

A quadrilateral can be drawn, if the measures of its

A
four sides are given

B
three sides and a diagonal are given

C
four sides and one angle are given

D
four angles and one side are given

Solution

For drawing a quadrilateral, minimum 5 elements are required in such a way that number of sides given must be greater than number of given angles. Only option which satisfies the the above condition is option C.
So, correct answer is option C.
Question 9
1 / -0

A Rhombus also satisfies the properties of a:

A
Square

B
Parallelogram

C
Trapezium

D
None of these

Solution

A rhombus is a parallelogram. The definition of a parallelogram is a quadrilateral that has $$2$$ pairs of parallel lines.

A parallelogram has properties including $$2$$ pairs of congruent sides, $$2$$ pairs of congruent opposite angles, supplementary consecutive angles, and diagonals that bisect each other.

A rhombus has all of these properties.
Question 10
1 / -0

Which of the following statement(s) is/are false?

A
Each diagonal of a quadrilateral divides it into two triangles

B
Each side of a quadrilateral is less than the sum of the remaining three sides

C
A quadrilateral can atmost have three obtuse angles

D
A quadrilateral has four diagonals

Solution

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Quadrilaterals Test - 18

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Quadrilaterals Test - 18

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Question 1

$$45^{\circ}$$

$$55^{\circ}$$

$$135^{\circ}$$

$$115^{\circ}$$

Solution

Question 2

$$75^o$$

$$90^o$$

$$105^o$$

$$100^o$$

Solution

Question 3

$$360^{\circ}$$

$$100^{\circ}$$

$$180^{\circ}$$

$$80^{\circ}$$

Solution

Question 4

A parallelogram having a pair of adjacent sides equal, is called a rhombus

The diagonals of a rhombus do not bisect each other

If the diagonals of a parallelogram bisect each other at right angles, it is a rhombus

A rectangle is a parallelogram

Solution

Question 5

$$140^o$$

$$120^o$$

$$180^o$$

$$220^o$$

Question 6

Rectangle

Rhombus

pentagon

Parallelogram

Solution

Question 7

$$2$$

$$3$$

$$4$$

$$360$$

Solution

Question 8

four sides are given

three sides and a diagonal are given

four sides and one angle are given

four angles and one side are given

Solution

Question 9

Square

Parallelogram

Trapezium

None of these

Solution

Question 10

Each diagonal of a quadrilateral divides it into two triangles

Each side of a quadrilateral is less than the sum of the remaining three sides

A quadrilateral can atmost have three obtuse angles

A quadrilateral has four diagonals

Solution

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