1. Construct a quadrilateral ABCD in which AB = 4.2 cm, BC = 3.6 cm, CD = 4.8 cm , AD = 5 cm and ∠B = 60^o

         Step 1 :-  Draw a line segment AB = 4.2 cm.

         Step 2 : At B,construct  ∠ABX = 60^o.

         Step 3 : From B, set off BC = 3.6 cm.

         Step 4 : With A as centre and radius equal to 5 cm,drawn an arc.

         Step 5 : With C as centre and radius equal to 4.8 cm, draw another arc, cutting the previous arc at D.

         Step 6 : Join AD and CD.

                                Then , ABCD is the required quadrilateral.

2. Construct a quadrilateral PQRS in which PQ = 6 cm, QR = 4 cm, RS = 3.5 cm and PS = 5 cm and ∠Q = 120^o

           Step 1 : Draw a line segment PQ = 6 cm.

       Step 2 : At Q, construct ∠PQX = 120^o .

       Step 3 : From X , set of R QR = 4 cm.

       Step 4 : With P as centre and radius equal to 5 cm draw an arc.

       Step 5 : With R as centre and radius equal to 3.5 cm draw another arc,cutting the previous arc at Q.

       Step 6 : Join PS and RS. Then, PQRS is the required quadrilateral.

3. Draw a rectangle of diagonal 7 cm and angle between the diagonal is 60^o.

      Method of construction

Draw AB = 7 cm.Find its mid point O by drawing its perpendicular bisector.Put the protractor on AC such that its middle point touches O and draw an angle of measure 50⁰.Extend this line to the other side AC.Mark B and D on this line so that OB = OD = OA.Join AB, BC, CD and AD to get the rectangle ABCD.

4. Construct a parallelogram one of whose sides is 5.2 cm and whose diagonals are 6 cm and 6.4 cm?

           Step 1 : Draw a line segment AB = 5.2 cm.

       Step 2 : With A as centre and radius 3.2 cm , draw an arc.

       Step 3 : With B as centre and radius 3 cm,draw another arc,cutting the previous arc at O.

       Step 4 : Join OA and OB.

       Step 5 : Produce AO to C such that OC = AO and produce BO to D such that OD = BO.

       Step 6 : Join AD , BC and CD.

Then ABCD is the required parallelogram.

5. Construct a rectangle ABCD in which AB = 5 cm and BC = 4 cm.

            Step 1 : Draw a line segment AB of length 5 cm.

        Step 2 : At B, draw BE ⊥ AB.

        Step 3 : With B as centre and radius 4 cm, draw an arc, cutting BE at C.

        Step 4 : With A as centre and radius 4 cm, draw an arc.

        Step 5 : With C as centre and radius 5 cm, draw another arc,cutting the previous arc at D.

        Step 6 : Join AD and CD.

Then , ABCD is the required rectangle

6. Draw a square ABCD in which side AB = 4.5 cm?

        Step 1 : - Draw a line segment AB = 4.5 cm.

        Step 2 : Draw B X ⊥ AB

        Step 3 : With B as centre and radius 4.5 cm, draw an arc , cutting BX at C

        Step 4 :  With A as centre and radius 4.5 cm, draw an arc

        Step 5 : With C as centre and the same radius,draw another arc , cutting the previous arc at D

        Step 6 : Join AD and CD

             Then ABCD is the required square

7. Draw a square ABCD in which diagonal AC = 5.4 cm?

       Step 1 : - Draw a line segment AC = 5.4 cm.

       Step 2 : -  Draw the right bisector XY of AC , meeting AC at O.

       Step 3 : - With centre O and radius OC, draw arcs , cutting OY at B and OX at D.

       Step 4 : - Join AB,BC,CD and DA.

Then , ABCD is the required square.

8. Explain the properties of a rectangle?

a) Opposite sides are of equal length.

b)  Opposite sides are parallel.

c) All angles are right angles.

9. Explain the properties of a square?

a) All sides are of equal length.

b) All angles are right angles.

10. Explain the properties of a parallelogram.

a) Opposite sides are of same length.

b) Opposite sides are parallel.

c) Opposite angles are equal.

11. Explain the properties of a Rhombus?

a) All sides are equal.

b) Opposite sides are parallel.

c) Opposite angles are equal.

12. Explain adjacent sides and opposite sides of a quadrilateral?

A quadrilateral has four angles as well as four sides.Sides of a quadrilateral that meet at one vertex (corner) are adjacent sides and sides that are not like this are called opposite sides.

13. Prove that the diagonals of an isosceles trapezium are equal.

ABCD is an isosceles trapezium. AB || CD and AD = BC

Consider ΔABC and ΔBAD.AB is common side.

AD = BC

As the angles of isosceles trapeziums on parallel sides are equal ∠CBA = ∠DAB

 As two sides and included angle of one triangle are equal to two sides and the angle determined by the sides of another triangle Δ ABC ≅Δ BAD.From the congruence of triangle , AC = BD

14. Prove that if the diagonals of a quadrilateral bisect each other,then it is a parallelogram.

AC and BD are the diagonals of quadrilateral ABCD.

So AO = OC , OD = OB

In ΔAOB and ΔCOD

AO = CO

OB = OD and 

∠AOB = ∠COD (Opposite angles)

Two sides and included angle of one triangle are equal to two sides and included angle of another triangle.So ∠ AOB  ∠COD.

15. Can we say that if one pair of opposite angles of a trapezium are made right angles then it becomes a rectangle?

ABCD is a trapezium , AB || CD

Also ∠A = 90^o and ∠C = 90^o

Since AB || CD

∠ A + ∠D = 180^o

90 + ∠D = 180^o

∠ D = 180 - 90 = 90^o

Since AB || CD, ∠C + ∠B = 180^o

90^o +∠ B = 180^o

∠ B = 180 - 90 = 90^o

As ∠A = ∠B = ∠C =∠ D = 90^o

ABCD is a rectangle.                                         

16. Each side of a rhombus is 12 centimetres long and one of its diagonals is 20 centimetres long.How long is the other diagonal?

As the diagonals of a rhombus are perpendicular bisectors to each other.

∠AOB = 90^o

In right angled triangle AOB ;

AB = 12 cm

AO = 10 cm

OB² = AB² - AO²

        = 12² - 10²

        = 144 - 100 = 44

  OB = √44

            =  = 2 √11

  DB = 2 ×2 √11

        = 4 ×11

17. The diagonals of a rhombus are of lengths 16 centimetres and 12 centimetres. What is its perimeter?

In right angled  ΔMQN

MQ = 8 cm

QN = 6 cm , and ∠MQN = 90^o

MN² = MQ² + QN² = 8² + 6² = 64 + 36 = 100

Side , MN = 100 = 10 cm

Perimeter = 4 ×10 cm = 40 cm.

18. Draw the quadrilaterals shown below.

3)

1)  Draw a line of 5 cm length. Left and of this line make an angle 80^o, a line of 3 cm is drawn. The other end of this line draw a line of 4 cm making an angle 120^o. Join the end points of the first line and the line latest drawn.

2) Draw a line of 5 cm long. At one end draw a line of 3 cm with an angle 60^o. At the other end draw a line making an angle 80^o. The line of 3 cm length make an angle 100^o and draw a line. The  point of intersection of this line and the line with angle 80^o is the fourth vertex.

3) Draw triangle of sides 7 cm, 8 cm and 4 cm. At the end points of line of 4 cm draw two circle parts of radius 5 cm and 6 cm. Join this point and the other end of the line of 7 cm. Hence the quadrilateral completed.