Understanding Quadrilaterals Test

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Understanding Quadrilaterals Test - 24

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Question 1
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Suppose $$ABCD$$ is a parallelogram in which $$\angle A = 108^{\circ}$$. Calculate $$\angle B, \angle C$$ and $$\angle D$$.

A
$$72^{\circ}, 108^{\circ}, 72^{\circ}$$

B
$$22^{\circ}, 108^{\circ}, 72^{\circ}$$

C
$$108^{\circ}, 72^{\circ}, 108^{\circ}$$

D
None of these

Solution

$$\Rightarrow$$ $$ABCD$$ is a parallelogram.
$$\Rightarrow$$ $$\angle A=108^o$$ [ Given ]
$$\Rightarrow$$ $$\angle A=\angle C$$ [ Opposite angles of a parallelogram are equal ]
$$\therefore$$ $$\angle C=108^o$$
$$\Rightarrow$$ $$\angle A+\angle D=180^o$$ [ Adjacent angles of parallelogram are supplementary ]
$$\Rightarrow$$ $$108^o+\angle D=180^o$$
$$\therefore$$ $$\angle D=72^o$$
$$\Rightarrow$$ $$\angle D=\angle B$$ [ Opposite angles of a parallelogram are equal ]
$$\therefore$$ $$\angle B=72^o$$
Question 2
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Which of the following statement(s) is/are correct?

A
A parallelogram in which two adjacent angles are equal is a rectangle.

B
A quadrilateral in which both pairs of opposite angles are equal is a parallelogram.

C
In a parallelogram, the maximum number of acute angles can be two.

D
All of these

Solution

$$\Rightarrow$$ We know that, rectangle is a parallelogram and all four angles of rectangle is right angle, so its two adjacent angles are equal.
$$\Rightarrow$$ A quadrilateral in which both pairs of opposite angles are equal is a parallelogram.
$$\Rightarrow$$ In a parallelogram, adjacent angles sum to $$180^o$$. Thus, there're $$2$$ acute and $$2$$ obtuse angles in a parallelogram.
$$\therefore$$ All the statements are correct in given options.
Question 3
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The angles A, B, C, D in the parallelogram ABCD are ___________.

A
$$90^o, 90^o, 90^o, 90^o$$

B
$$90^o, 45^o, 30^o, 15^o$$

C
$$30^o, 60^o, 20^o, 40^o$$

D
$$90^o, 45^o, 15^o, 20^o$$

Solution

In parallelogram $$ABCD,\, AB ||DC $$
$$\therefore \angle ADB=\angle CBD$$
and $$\angle ABD=\angle BDC$$
$$\Rightarrow \angle ABD=2a\\ \Rightarrow \angle ADB=3a$$
$$\therefore \angle B=2a+3a=5a$$
and $$\angle D=3a+2a=5a $$
In $$\triangle BCD: $$
$$3a+5a+2a=180^o\\ \Rightarrow a=18$$
$$\therefore \angle B=\angle C=\angle D=90^o$$
For $$\angle A :$$
$$\angle A=360-\angle B-\angle D-\angle C\\=180-90-90-90\\=90^o$$
Question 4
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Which of the following statements is definitely true, if PQRS is a parallelogram?

A
$$PQ$$ and $$SR$$ are adjacent sides.

B
$$\angle P$$ and $$\angle Q$$ are opposite angles

C
$$SR \parallel PQ$$

D
$$PR=RQ$$

Solution

Given: $$PQRS$$ is a parallelogram.

We can draw it in $$2$$ possible ways.

So, $$PQ$$ and $$SR$$ may or may not be adjacent sides.

and $$SR$$ is parallel to $$PQ$$ in either case.

Also, $$\angle P$$ and $$\angle Q$$ are adjacent angles as they lie on the same edge.

Sides $$PR$$ is not equal to $$RQ$$ as one of them is side and other is the diagnol.
Question 5
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ABCD is a parallelogram and L is a point on DB. The produced line AL meets BC at M and DC produced at N. Given that DL$$=3$$LB, find $$\displaystyle\frac{AB}{CN}$$.

A
$$3/2$$

B
$$1/2$$

C
$$4/5$$

D
$$1/4$$

Solution

In $$\triangle LAB $$ and $$ \triangle DLN $$ ,
$$ \angle ALB = \angle DLN $$ are vertically opposite and $$ \angle LAB = \angle DNL $$ (alternate angles) and third angle is also equal.
So both triangles are similar.
Therefore $$ \dfrac{DN}{AB} = \dfrac{DL}{LB} $$
$$ DN=DC+CN $$
So, $$1+\dfrac{CN}{AB}=3 $$
$$ \dfrac{AB}{CN}=\dfrac{1}{2}$$
Hence option B is correct.
Question 6
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Select the INCORRECT match.

A
Quadrilateral with one pair of parallel sides is a Trapezium

B
A rhombus with $$4$$ right angles is a Square

C
Parallelogram with $$4$$ right angles ia a Rectangle

D
All sides of a quadrilateral are equal then it is a Rectangle

Solution

$$\Rightarrow$$ We know that, in Trapezium there is only one pair of parallel side.

$$\Rightarrow$$ Rhombus has its all four sides equal and if its all angle four becomes right angle then its called square.

$$\Rightarrow$$ In rectangle pair of opposite sides are equal, so rectangle is also a parallelogram with $$4$$ right angle.

$$\therefore$$ Option $$A,B,C$$ are correct.

$$\Rightarrow$$ Option $$D$$ is incorrect.
Question 7
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Choose the correct answer from the alternatives given.
In a parallelogram $$PQRS,$$ angle $$P$$ is four times of angle $$Q.$$ then the measure of $$\angle R$$ is:

A
$$36^{\circ}$$

B
$$72^{\circ}$$

C
$$130^{\circ}$$

D
$$144^{\circ}$$

Solution

Given:
$$\angle P \, = \, 4 \angle Q$$
Let $$\angle Q =x$$
$$\therefore \, \angle P \, = \, 4x$$
$$\angle P \, + \, \angle Q \, = \, 180^{\circ}$$
$$4x \, + \, x \, = \, 180^{\circ}$$
$$5x=180^o$$
$$\Rightarrow x \, = \, 36^{\circ}$$
Thus, $$\angle P=4\times 36^o$$
$$\Rightarrow \angle P=144^o$$
In parallelogram opposite sides are equal, thus $$\angle P=\angle R$$
$$\Rightarrow \angle R=144^o $$
Question 8
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If $$ABCD$$ is a parallelogram whose diagonals intersect at $$O$$ and $$\triangle BCD $$ is an equilateral triangle having each side of length $$6$$ cm, then the length of diagonal $$AC$$ is:

A
$$2\sqrt3$$ cm

B
$$6\sqrt3$$ cm

C
$$3\sqrt6$$ cm

D
$$12 $$ cm

Solution

In $$\triangle ADB$$,
$$ DA = AB = DB = 6\ \text{cm}$$ (Given)

Altitude of $$\triangle ADB = OA$$

$$ = \dfrac{\sqrt3}{2} \times 6 $$

$$= 3\sqrt 3$$

$$\therefore AC = 2.OA$$

$$=2(3\sqrt3)\ \text{cm}$$

$$= 6\sqrt3\ \text{cm} $$
Question 9
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In a parallelogram ABCD, $$\angle B= (2x+ 25)^0$$ and $$\angle D= ( 3x-5)^0$$. Find:
(i) The value of $$x$$
(ii) Measure of angle.

A
x=30, b=91 degrees, d=89 degrees

B
x=32, b=91 degrees, d=89 degrees

C
x=30, b=89 degrees, d=91 degrees

D
x=32, b=89 degrees, d=91 degrees

Solution

In $$\parallel gm$$ opposite angles are supplementary.
Therefore,
$$\angle B+\angle D=180^{\circ}$$
$$2x+25+3x-5=180^{\circ}$$
$$x=\frac{180-20}{5}=32$$
$$\angle B=2x+25=(2*32)+25=89^{\circ}$$
$$\angle D=3x-5=(3*32)-5=91^{\circ}$$
Question 10
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An exterior angle and an interior angle of a regular polygon are in the ratio $$2: 7$$. Find the number of sides in the polygon.

A
$$2$$

B
$$7$$

C
$$9$$

D
$$18$$

Solution

Exterior angle $$=\dfrac {360^{\circ}}{n}$$ (n is number of sides of a polygon)
Interior angle $$=\dfrac {(n-2)180^{\circ}}{n}$$

$$\Rightarrow \dfrac {2}{7}=\dfrac {360^{\circ}}{(n-2)180^{\circ}}$$

$$\Rightarrow n-2=7$$

$$\Rightarrow n=9$$

Hence, the number of sides in the polygon is $$9$$.