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

8th Grade > Mathematics

UNDERSTANDING QUADRILATERALS MCQs

Total Questions : 464 | Page 41 of 47 pages

Question 143
A playground in the town is in the form of a kite. The perimeter is 106 m. If one of its sides is 23 m, what are the lengths of other three sides?

Discuss Question

Answer: Option A. ->
:
In a kite, two pairs of adjacent sides are equal. Since we know that one side measures 23 m in length, we can assume that the side adjacent to it will also be of the same length.
Let the length of the opposite side be x m. Then, the length of the side adjacent to this side will also be x m.
Perimeter of the playground =

106 m

\Rightarrow 23 + 23 + x + x = 106 \Rightarrow 46 + 2 x = 106 \Rightarrow 2 x = 106 - 46 \Rightarrow 2 x = 60 \Rightarrow x = 30 m

Hence, the lengths of the other three sides are 23m, 30m and 30m.

Question 402.

Question 145
In rectangle PAIR, find $∠ A R I, ∠ R M I$ and $∠ P M A$ .

Discuss Question

Answer: Option A. ->
:
Given,

∠ R A I = 35^{\circ}

∴ ∠ P R A = 35^{\circ}

. [

P R ∥ A I

and AR is transversal]

\Rightarrow ∠ A R I = 90^{\circ} - ∠ P R A = 90^{\circ} - 35^{\circ} = 55^{\circ}

∵ R M = I M, ∠ M R I = ∠ M I R = 55^{\circ}

.

In

Δ R M I,

∠ R M I + ∠ M R I + ∠ M I R = 180^{\circ}

. [Angle sum property]

\Rightarrow ∠ R M I = 180^{\circ} - 55^{\circ} - 55^{\circ} = 180^{\circ} - 110^{\circ} = 70^{\circ}

Also,

∠ P M A = ∠ R M I

[vertically opposite angles]

\Rightarrow ∠ P M A = 70^{\circ}

Question 403.

Question 146
In parallelogram ABCD, find $∠ B, ∠ C$ and $∠ D$ .

Discuss Question

Answer: Option A. ->
:
In a parallelogram, the opposite angles are equal, therefore

∠ C = ∠ A = 80^{\circ}

.
Also, adjacent angles are supplementary.

∴ ∠ A + ∠ B = 180^{\circ} 80^{\circ} + ∠ B = 180^{\circ} ∠ B = 180^{\circ} - 80^{\circ} \Rightarrow ∠ B = 100^{\circ}

Now,

∠ B = ∠ D

[Opposite angles are equal]

∴ ∠ D = 100^{\circ}

Question 404.

Question 147
In parallelogram PQRS, O is the mid-point of SQ. Find $∠ S, ∠ R$ , PQ, QR and diagonal PR.

Discuss Question

Answer: Option A. ->
:
Given,

∠ R Q Y = 60^{\circ}

∠ S R Q = ∠ R Q Y

[

S R ∥ P Y

, RQ is the transversal]

\Rightarrow ∠ R = 60^{\circ}

∴ ∠ S = 180^{\circ} - ∠ R = 180^{\circ} - 60^{\circ} = 120^{\circ}

[

∵

adjacent angles are supplementary]
Also, SR = 15 cm

\Rightarrow

PQ = 15 cm [

∵

opposite sides of a parallelogram are equal]
And PS = 11 cm

\Rightarrow

QR = 11 cm [

∵

opposite sides of a parallelogram are equal]
PR is bisected by SQ, so

P R = 2 \times P O = 2 \times 6 = 12 c m

Question 405.

Question 148
In rhombus BEAM, find $∠ A M E$ and $∠ A E M$ .

Discuss Question

Answer: Option A. ->
:
Given,

∠ B A M = 70^{\circ}

.
We know that, in rhombus, diagonals bisect each other at right angles.

∴ ∠ B O M = ∠ B O E = ∠ A O M = ∠ A O E = 90^{\circ}

.
Now, in

Δ A O M

∠ A O M + ∠ A M O + ∠ O A M = 180^{\circ}

. [angle sum property of triangle]

\Rightarrow 90^{\circ} + ∠ A M O + 70^{\circ} = 180^{\circ} \Rightarrow ∠ A M O = 180^{\circ} - 90^{\circ} - 70^{\circ} \Rightarrow ∠ A M O = 20^{\circ}

Also, AM = BM = BE = EA.
In

Δ A M E

, we have,
AM = EA

∴ ∠ A M E = ∠ A E M = 20^{\circ}

[

∵

equal sides make equal angles]

Question 406.

Question 149
In parallelogram FIST, find $∠ S F T, ∠ O S T$ and $∠ S T O$ .

Discuss Question

Answer: Option A. ->
:
Given,

∠ F I S = 60^{\circ}

Now,

∠ F T S = ∠ F I S = 60^{\circ}

[

∵

opposite angles of a parallelogram are equal]
Now,

F T ∥ I S

and TI is a transversal, therefore

∠ F I O = ∠ S T O = 35^{\circ}

[alternate angles]
Also,

∠ F O T + ∠ S O T = 180^{\circ}

[linear pair]

\Rightarrow 110^{\circ} + ∠ S O T = 180^{\circ}

\Rightarrow ∠ S O T = 70^{\circ}

Δ T O S, ∠ T S O + ∠ O T S + ∠ T O S = 180^{\circ}

[angle sum property of triangle]

∴ ∠ O S T = 180^{\circ} - (70^{\circ} + 35^{\circ}) = 75^{\circ}

Δ F S T, ∠ S F T + ∠ F T S + ∠ T S F = 180^{\circ}

[angle sum property of triangle]

\Rightarrow ∠ S F T = 180^{\circ} - (60^{\circ} + 75^{\circ}) ∴ ∠ S F T = 45^{\circ}

Question 407.

Question 150
In the given parallelogram YOUR, $∠ R U O = 120^{\circ}$ and OY is extended to point S, such that $∠ S R Y = 50^{\circ}$ . Find $∠ Y S R$ .