Quadrilaterals Test

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Quadrilaterals Test - 7

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Question 1
1 / -0

In quadrilateral ABCD, if ∠A = 60^∘ and ∠B : ∠C : ∠D = 2:3:7, then ∠D is :

A
50^∘

B
175^∘

C
180^∘

D
25^∘

Solution

In quadrilateral, the sum of the all four angles equal to 360. let angle B = 2x, angle C = 3x and angle D = 7x.

angle A + angle B + angle C + angle D = 360

60 + 2x + 3x + 7x = 360

12x = 300

x= 25

So, angle D = 7x = 7 (25) = 175
Question 2
1 / -0

If APB and CQD are 2 parallel lines, then the bisectors of the angles APQ, BPQ, CQP and PQD form, square only if

A
Diagonals of ABCD are equal

B
Diagonals of ABCD are unequal

C
ABCD is a Rhombu

D
None of these

Solution

The diagonals of a square bisect its angles. Opposite sides of a square are both parallel and equal in length. All four angles of a square are equal.
Question 3
1 / -0

The figure formed by joining the mid-points of the sides of a quadrilateral ABCD, taken in order, is a square only if

A
Diagonals of ABCD are equal and perpendicular

B
ABCD is a Rhombus

C
Diagonals of ABCD are equal

D
Diagonals of ABCD are perpendicular

Solution

A quadrilateral formed by joining the mid points of a square is a square. So, ABCD is a square. In Square, diagonals are equal and perpendicular.
Question 4
1 / -0

D and E are the mid-points of the sides AB and AC of △ABC and O is any point on the side BC, O is joined to A. If P and Q are the mid-points of OB and OC res, Then DEQP is

A
A Parallelogram

B
A Triangle

C
A Rhombus

D
A Rectangle

Solution

By mid point theorem, DE is parallel to BC. In triangle BOA, DP parallel to OA and OA is parallel to QE in triangle AOC( mid point theorem) because D and P are mid points in triangle BOA and E and Q are mid points in triangle AOC.

So, DP is parallel to EQ. In quadrilateral DPQE, both pair of opposite sides are parallel. So, it become parallelogram.
Question 5
1 / -0

The bisectors of the angles of a Parallelogram enclose a

A
Rhombus

B
Square

C
Rectangle

D
Parallelogram

Solution

Let ABCD be a parallelogram. angle A + angle D = 180 ( co-interior angles)

1/2 of (angle A + angle D) = 90.

Triangle formed by bisectors of angle A and angle D, have sum of two angles equals to 90 therefore, remaining angle is of 90. similarly we can prove that other angles formed are of 90 each by bisectors of angles of ABCD. The quadrilateral formed by angle bisectors of ABCD has all angles equal to 90 ( Vertically opposite angles). A quadrilateral with all right angles is a Rectangle.
Question 6
1 / -0

Rhombus is a quadrilateral

A
in which diagonals are inclines at an angle of 60^∘

B
in which diagonals are equal

C
in which diagonals are inclines at an angle of 120.

D
in which diagonals bisect opposite angles

Solution

Let ABCD be a rhombus.

Join BD which forms two triangles ABD and DCB. In triangle ABD, AB = AD.

So, angle ADB = angle ABD ( angles opposite to equal sides are equal) ----------(1)

But , angle ABD = angle BDC and angle ADB = angle CBD ( alternate angles)-------------(2)

So, from (1) and (2)

angle ADB = angle ABD = angle BDC = angle CBD

therefore, diagonal BD bisects the angle B and angle D.
Question 7
1 / -0

In Quadrilateral ABCD, ∠A= 110^∘, ∠B = 75^∘ and ∠C = 35⁰. Find ∠D?

A
145^∘

B
110^∘

C
50^∘

D
140^∘

Solution

angle A + angle B + angle C + angle D = 360 ( angle sum property of quadrilateral)

110 + 75 + 35 + angle D = 360

angle D = 360-220 = 140
Question 8
1 / -0

In Quadrilateral ABCD, ∠A = (3x) ^∘, ∠B = (5x) ^∘, ∠C = (20x) ^∘, ∠D = (8x) ^∘. Find the value of x?

A
10

B
9

C
20

D
11

Solution

angle A + angle B + angle C + angle D = 360 ( angle sum property)

3x + 5x + 20x + 8x = 360

36x = 360

x = 10
Question 9
1 / -0

In Quadrilateral ∠A = 38^∘, ∠C = 3 ∠A, ∠D = 4∠A. Find the value of ∠B?

A
80^∘

B
57^∘

C
56^∘

D
55^∘

Solution

angle A + angle B + angle C + angle D = 360 ( angle sum property)

angle C = 3 ( 38) = 114

angle D = 4 ( 38) = 152

So, 38 + angle B + 114 + 152 = 360

angle B = 360 - 304 = 56

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Quadrilaterals Test - 7

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Quadrilaterals Test - 7

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SHARING IS CARING

Question 1

50∘

175∘

180∘

25∘

Solution

Question 2

Diagonals of ABCD are equal

Diagonals of ABCD are unequal

ABCD is a Rhombu

None of these

Solution

Question 3

Diagonals of ABCD are equal and perpendicular

ABCD is a Rhombus

Diagonals of ABCD are equal

Diagonals of ABCD are perpendicular

Solution

Question 4

A Parallelogram

A Triangle

A Rhombus

A Rectangle

Solution

Question 5

Rhombus

Square

Rectangle

Parallelogram

Solution

Question 6

in which diagonals are inclines at an angle of 60∘

in which diagonals are equal

in which diagonals are inclines at an angle of 120.

in which diagonals bisect opposite angles

Solution

Question 7

145∘

110∘

50∘

140∘

Solution

Question 8

10

9

20

11

Solution

Question 9

80∘

57∘

56∘

55∘

Solution

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50^∘

175^∘

180^∘

25^∘

in which diagonals are inclines at an angle of 60^∘

145^∘

110^∘

50^∘

140^∘

80^∘

57^∘

56^∘

55^∘