Understanding Quadrilaterals Test

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Understanding Quadrilaterals Test - 7

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Question 1
1 / -0

If an angle of a parallelogram is two-third of its adjacent angle, then what is the smallest angle of parallelogram?

A
$$108^{\circ}$$

B
$$54^{\circ}$$

C
$$72^{\circ}$$

D
$$81^{\circ}$$

Solution

Let one angle be $$\angle x$$
Thus, smallest angle is $$\dfrac{2}{3}\angle x$$
Now, $$\dfrac{2}{3}\angle x + \angle x =180$$

$$\Rightarrow \dfrac{2+3}{3}\angle x=180$$

$$\Rightarrow \angle x =\dfrac{180\times 3}{5}$$

$$\Rightarrow \angle x=108$$
Thus, the smallest angle is $$\dfrac{2}{3}\times 108=72^o$$
Question 2
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In a parallelogram ABCD(naming is done in counter - clockwise direction), determine the sum of angle $$\angle$$C and $$\angle$$D.

A
$$\angle$$C + $$\angle$$D = 180$$^o$$

B
$$\angle$$C + $$\angle$$D = 270$$^o$$

C
$$\angle$$C + $$\angle$$D = 90$$^o$$

D
$$\angle$$C + $$\angle$$D = 360$$^o$$

Solution

$$ABCD$$ is a parallelogram,
$$AD\parallel BC$$ and $$CD$$ is the transversal.
Thus, $$\angle C +\angle D=180^{o}$$ [Sum of the internal angles on the same side of the transversal is 180]
Question 3
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Which of the following pair of shapes when joined together (by placing from edge to edge) can form a rectangle?

A

B

C

D
None of these

Solution

Properties of rectangle are :
$$\Rightarrow$$ Each of the interior angles is 90$$^\circ$$.
$$\Rightarrow$$ Opposite sides of rectangle are parallel.
$$\Rightarrow$$ Opposite sides of rectangle are equal.
$$\Rightarrow$$ So, by joining pair of shapes of option B and C we will not form rectangle because they are not satisfying all the properties of rectangle.
$$\Rightarrow$$ So, only joining pair of shapes of option A, can form a rectangle because it's satisfy all the properties of rectangle.
Question 4
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If one of the angle measures more than $$180^{\circ}$$ in a quadrilateral, then that is known as ___ .

A
a parallelogram

B
a concave quadrilateral

C
a convex quadrilateral

D
a trapezium

Solution

If one of the angle measures more than $$ {180}^{o} $$ in a quadrilateral, then that is known as a concave quadrilateral.
Question 5
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In a parallelogram ABCD, if $$\angle$$B = 135$$^\circ$$, determine the measures of its other angles

A
$$\angle A =\angle C = 45^\circ,\ \angle B = \angle D = 135^\circ$$

B
$$\angle A = \angle C = 35^\circ,\ \angle B = \angle D = 135^\circ$$

C
$$\angle A = \angle C = 55^\circ,\ \angle B = \angle D = 135^\circ$$

D
None of the above

Solution

Given: In $$ABCD,$$ $$\angle B=135^\circ$$

We know that adjacent angles of a parallelogram are supplementary

So, $$\angle A+\angle B=180^\circ$$

$$\Rightarrow \angle A=180-135$$

$$\Rightarrow \angle A=45^\circ$$

And the opposite angles of a parallelogram are equal.

Hence $$\angle B=\angle D=135^\circ$$

And $$\angle A=\angle C=45^\circ$$
Question 6
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Number of pairs of parallel lines in a trapezium is:

A
$$1$$

B
$$2$$

C
$$3$$

D
$$4$$

Solution

A trapezium has one pair of parallel lines as shown in the figure. So option $$A$$ is correct.
Question 7
1 / -0

Given here are some figures:
Which one is concave polygon among $$1$$ and $$2$$?

A
$$1$$

B
$$2$$

C
Both

D
None of these

Solution

A polygon that has one or more interior angles greater than 180 known as concave polygon.
Question 8
1 / -0

Explain how square is a parallelogram.

A
sum of all angles is $$90^o$$

B
All angles are equal

C
Opposites sides are parallel

D
None of these

Solution

$$\Rightarrow$$ A parallelogram is a quadrilateral with two pairs of opposite sides.
$$\Rightarrow$$ In square we know that, all sides are equal which means opposite sides are equal.
$$\Rightarrow$$ And in square opposite sides are parallel.
Question 9
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$$AB=DC=8 cm, AD=4 cm$$ what should be the length of side $$BC$$, if $$ABCD$$ is parallelogram

A
$$4cm$$

B
$$8cm$$

C
$$16cm$$

D
None of the above

Solution

As $$ABCD$$ is parallelogram so opposite sides should be equal
$$\Rightarrow AD=BC$$
$$\therefore BC=4cm$$
Question 10
1 / -0

The diagonals of a parallelogram ABCD intersect at O. If $$\angle BOC =90^o$$ and $$\angle BDC = 50^o$$, then $$\angle AOB$$ is

A
$$10^o$$

B
$$40^o$$

C
$$50^o$$

D
$$90^o$$

Solution

It is given that angle BOC = 90.
It means the diagonals cut each other at a right angle.
Angle AOB= 180- Angle BOC =90
So, angle AOB = angle BOC = 90.
So the correct answer is option D.
The data that $$\angle BDC = 50^{\circ}$$ is not required.