Understanding Quadrilaterals(8th Grade > Mathematics ) Questions and answers for exam Preparation

8th Grade > Mathematics

UNDERSTANDING QUADRILATERALS MCQs

Total Questions : 464 | Page 44 of 47 pages

Question 169
In the figure, find the value of x.

Discuss Question

Answer: Option A. ->
:
We know that, sum of all the exterior angles of a polygon is

360^{\circ}

∴ 92^{\circ} + 20^{\circ} + 85^{\circ} + x^{\circ} + 89^{\circ} = 360^{\circ} \Rightarrow 286^{\circ} + x^{\circ} = 360^{\circ} \Rightarrow x^{\circ} = 360^{\circ} - 286^{\circ} = 74^{\circ}

Question 432.

Question 171
In a quadrilateral HOPE, PS and ES are bisectors of $∠ P$ and $∠ E$ respectively. Give reason.

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Answer: Option A. ->
:
Data insufficient

Question 433.

Question 172
ABCD is a parallelogram. Find the values of x, y and z.

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Answer: Option A. ->
:
Given, a parallelogram ABCD
In the

Δ O B C

, we have

y + 30^{\circ} = 100^{\circ}

[exterior angle property of triangle]

\Rightarrow y = 70^{\circ}

By the angle sum property of a triangle,
we have,

x + y + 30 = 180^{\circ}

\Rightarrow x + 70^{\circ} + 30^{\circ} = 180^{\circ} \Rightarrow x = 180^{\circ} - 100^{\circ} = 80^{\circ}

Now, since

A D ∥ B C

and BD is transversal, therefore

∠ A D O = ∠ O B C

[alternate interior angles]

\Rightarrow z = 30^{\circ}

Question 434.

Question 174
ABCD is a trapezium such that $A B ∥ C D$ , $∠ A : ∠ D = 2 : 1, ∠ B : ∠ C = 7 : 5$ . Find the angles of the trapezium.

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Answer: Option A. ->
:
Let ABCD be a trapezium, where

A B ∥ C D

Question 174ABCD Is A Trapezium Such That AB∥CD, ∠A:∠D...

Let the angles A and D be of measures 2x and x, respectively.
Then,

2 x + x = 180^{\circ}

[

∵

in trapezium, the angles on either side of the base are supplementary]

\Rightarrow 3 x = 180^{\circ} \Rightarrow x = 60^{\circ}

∴ ∠ A = 2 \times 60^{\circ} = 120^{\circ}, ∠ D = 60^{\circ}

Again, let the angles B and C be 7x and 5x respectively. Then,

7 x + 5 x = 180^{\circ}

\Rightarrow 12 x = 180^{\circ} \Rightarrow x = 15^{\circ}

Thus,

∠ B = 7 \times 15 = 105^{\circ}

and

∠ C = 5 \times 15 = 75^{\circ}

Question 435.

Question 173
Diagonals of a quadrilateral are perpendicular to each other. Is such a quadrilateral always a rhombus? Give a figure to justify your answer.

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Answer: Option A. ->
:
False, it is not necessary that a quadrilateral having perpendicular diagonals is a rhombus.
e.g. Consider a trapezium ABCD in which

A B ∥ C D

Question 173Diagonals Of A Quadrilateral Are Perpendicular T...

Question 436.

Question 175
A line l is parallel to line m and a transversal p intersects them at X, Y respectively. Bisectors of interior angles at X and Y intersect at P and Q. Is PXQY a rectangle? Give reason.

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Answer: Option A. ->
:
Given,

l ∥ m

∴ ∠ D X Y = ∠ X Y A

[alternate interior angles]

\Rightarrow \frac{∠ D X Y}{2} = \frac{∠ X Y A}{2}

[dividing both the sides by 2]

Question 175A Line L Is Parallel To Line M And A Transversal...

Now,

∠ 1 = ∠ 2

[

∵

XP and YQ are bisectors]
Alternate angles are equal, i.e.

∠ 1 = ∠ 2

\Rightarrow X P ∥ Q Y

....... (i)
Similarly,

X Q ∥ P Y

....... (ii)
From Eqs. (i) and (ii), we get
PXQY is a parallelogram ........ (iii)

\Rightarrow ∠ D X Y + ∠ X Y B = 180^{\circ}

[

∵

interior angles on the same side of transversal are supplementary]

\frac{∠ D X Y}{2} + \frac{X Y B}{2} = \frac{180^{\circ}}{2}

[dividing both the sides by 2]

∠ 1 + ∠ 3 = 90^{\circ}

....... (iv)
In

Δ X Y P

∠ 1 + ∠ 3 + ∠ P = 180^{\circ}

90^{\circ} + ∠ P = 180^{\circ}

[from Eq. (iv)]

∠ P = 90^{\circ}

......... (v)
Similarly,

∠ Q = 90^{\circ}

......... (vi)
From Eqs. (iii), (v) and (vi), PXQY is a rectangle.

Question 437.

Question 176
ABCD is a parallelogram. The bisector of angle A intersects CD at X and bisector of angle C intersects AB at Y. Is AXCY a parallelogram? Give reason.

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Answer: Option A. ->
:
Given, ABCD is a parallelogram.
So,

∠ A = ∠ C

[

∵

opposite angles of a parallelogram are equal]

Question 176ABCD Is A Parallelogram. The Bisector Of Angle A...

∴ \frac{∠ A}{2} = \frac{∠ C}{2}

[dividing both the sides by 2]

∠ 1 = ∠ 2

But

∠ 2 = ∠ 3

[

∵ A B ∥ C D

and CY is the transversal]

∴ ∠ 1 = ∠ 3

But they are pair of corresponding angles.

∴ A X ∥ Y C

..... (i)

A Y ∥ X C

[

∵ A B ∥ D C

] ...... (ii)
From Eqs. (i) and (ii), we get
AXCY is a parallelogram.

Question 438.

Question 178
The angle between the two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is $45^{\circ}$ . Find the angles of the parallelogram.

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Answer: Option A. ->
:
Let ABCD be a parallelogram, where BE and BF are the perpendicular through the vertex B to the sides DC and AD, respectively.

Question 178The Angle Between The Two Altitudes Of A Paralle...

Let

∠ A = ∠ C = x, ∠ B = ∠ D = y

[

∵

opposite angles are equal in parallelogram]
Now,

∠ A + ∠ B = 180^{\circ}

[

∵

adjacent sides of a parallelogram are supplementary]

\Rightarrow x + ∠ A B F + ∠ F B E + ∠ E B C = 180^{\circ}

\Rightarrow x + 90^{\circ} - x + 45^{\circ} + 90^{\circ} - x = 180^{\circ}

[∵ I n Δ A B F, ∠ A B F = 90^{\circ} - x a n d i n Δ B E C, ∠ E B C = 90^{\circ} - x]

\Rightarrow - x = 180^{\circ} - 225^{\circ} \Rightarrow x = 45^{\circ} ∴ ∠ A = ∠ C = 45^{\circ} ∠ B = 45^{\circ} + 45^{\circ} + 45^{\circ} = 135^{\circ} \Rightarrow ∠ D = 135^{\circ}

Hence the angles are

45^{\circ}, 135^{\circ}, 45^{\circ}, 135^{\circ}

Question 439.

Question 177
A diagonal of a parallelogram bisects an angle. Will it also bisect the other angle? Give reason.

Discuss Question

Answer: Option A. ->
:
Consider a parallelogram ABCD.
Given,

∠ 1 = ∠ 2

Since, ABCD is a parallelogram.

A B ∥ C D

and AC is the transversal.

∴ ∠ 1 = ∠ 4

[alternate angles] .... (i)
Similarly,

∠ 2 = ∠ 3

[alternate angles] .... (ii)
But given,

∠ 1 = ∠ 2

∴ ∠ 3 = ∠ 4

[from Eqs. (i) and (ii)]

Question 177A Diagonal Of A Parallelogram Bisects An Angle. ...

Yes, the diagonal will bisect the other angle.

Question 440.

Question 179
ABCD is a rhombus such that the perpendicular bisector of AB passes through D. Find the angles of the rhombus.
[Hint Join BD. Then, $Δ A B D$ is equilateral]