Which of the following is/are concave quadrilaterals?

A.

I & III

B.

II & IV

C.

IV

D.

II

#### SOLUTION

Solution : C

A line segment connecting any two non-consecutive vertices of a polygon is known as diagonal.A polygon with an exterior diagonal is known as concave polygon.

Only in these two options, there exists an exterior diagonal. So, these are concave polygons. But II is a concave pentagon. Only IV is a concave quadrilateral. Moreover, a concave polygon has one or more interior angle greater than 180°.

Which of the following is not true for the figure below?

A.

AC BD

B. ABCD is a kite
C.

D.

DAB = DCB

#### SOLUTION

Solution : D

It is given that AB = AD and BC = CD.
Since, two pairs of adjacent sides are equal, the ABCD is a kite.

According to the properties of a kite:

ACBD
[Diagonals of kite are perpendicular.]

In ΔABD,
ADB = ABD  [BA=DA and angles opposite to equal sides are equal]

But, DAB DCB

Which of the following quadrilaterals has/have equal length of diagonals?

i) Rhombus

ii)  Rectangle

iii) Square

iv) Kite

v) Parallelogram

A.

i, ii, iii

B.

ii, iii

C.

ii, iii, v

D.

ii, iii, iv

#### SOLUTION

Solution : B

Rhombus has unequal diagonals which are perpendicular bisectors.

Square has equal diagonals which are perpendicular bisectors.

Rectangle has equal diagonals which are perpendicular bisectors .

Kite has unequal diagonals in which the longer one bisects the other.

Parallelogram has unequal diagonals unless it is a square or a rectangle. The diagonals bisect each other.

Find ABF from the given figure.

A.

30

B.

45

C.

60

D.

75

#### SOLUTION

Solution : C

BCD+BCH=180  [ Linear pair]
BCD = 180 - 35 = 145
In quadrilateral BEDC, BED + EDC + DCB + CBE = 360
CBE = 360 - (105 + 50 + 145) = 60
So, ABF = CBE = 60(Vertically opposite angles)

ABCD is a parallelogram in which DCB and ABC are in ratio 2:7. Find the sum of ABD and ADB.

A.

40

B.

50

C.

70

D.

140

#### SOLUTION

Solution : D

DCB+ABC=180
[Adjacent angles of a parallelogram are supplementary]

2t+7t=180
[DCB and ABC are in ratio 2:7]

t=20

DCB=2t=2×20=40

and ABC=7t=7×20=140

Now, DAB=DCB=40
[Opposite angles of a parallelogram are equal]

[Sum of the angles of a triangle]

What is the measure of any exterior angle of a regular nonagon?

A.

40

B.

100

C.

140

D.

90

#### SOLUTION

Solution : A

Sum of the exterior angles of a polygon (irrespective of the number of sides) is 360.

In a nonagon, the measure of all exterior angles will be the same i.e. 360.

Also, nonagon has 9 sides.

So, measure of each exterior angle
= 3609=40

The total number of parallelograms in this figure is

___

#### SOLUTION

Solution :

Rectangles are also parallelograms. They are shown in color "blue" in the below figures. Hence, there are 7 parallelograms in the figure.

A.

True

B. False

#### SOLUTION

Solution : A

A polygon is called a regular polygon if it is equiangular and equilateral. i.e., All the angles are equal and measure of all the sides are equal.
Square is quadrilateral with all the angles equal to 90 and all the sides of equal length.
Hence, the given statement is true.

The diagonal of a parallelogram divides it into two congruent triangles.

A.

True

B.

False

#### SOLUTION

Solution : A

ΔABD and ΔCDB

AB = CD (Opposite sides of a parallelogram)

AD = CB (Opposite sides of a parallelogram)

BD = DB (Common)

So, ΔABD  ΔCDB      (By SSS congruence condition)

Select the correct statements.

A.

All rhombuses are parallelograms

B.

All kites are rhombuses

C. All rhombuses are kites
D. All parallelograms are kites

#### SOLUTION

Solution : A and C

Since, the opposite sides of a rhombus have the same length  it is also a parallelogram. So, a rhombus has all the properties of a parallelogram and also that of a kite. A quadrilateral with exactly two pairs of equal consecutive sides is a kite. This condition is also satisfied by Rhombus. So, all rhombuses are kites but converse is always not true