Quadrilateral1 pair of // opp. Sides

One of the diagionals is axis of symmetry2 diagionals are

2 pairs of equal adjacent sides

Sum of interior angles is 1800

2 pairs of opposite sides are equal.(opp. sides of // gram)2 pairs of opposite angles are equal (opp. s of // gram)Diagonals bisect each other (diag. Of // gram)

2 pairs of opp.// sides

4 right angles

Diagonals are equal

Properties of trapesium

Properties of // gram

Diagonals bisects each interior angle

KiteTrapezium

Rectangle

Rhombus4 equal sides

Properties of // gram and kite

Angles between each diagional and each side is 450

450

Properties of rhombus/rectangle

4 right angles and 4 equal sides

Parallelogram

Square

Trapeziums Definition : 1 pair of parallel sides

Properties:

Sum of interior angles is 1800

Parallelogram Definition : 2 pairs of opp. parallel sides

Properties:

2 pairs of opposite sides are equal.(opp. sides of // gram)

2 pairs of opposite angles are equal (opp. s of // gram)

Diagonals bisect each other(diag. Of // gram)

Conditions for Parallelogram

If 2 pairs of opposite angles are equal thenthe quadrilateral is parallelogram. (opp. s of eq.)

If diagonals bisect each other thenthe quadrilateral is parallelogram(diag. Bisect each other)

If 2 pairs of opposite sides are equal thenthe quadrilateral is parallelogram.(opp. sides eq.)

If 1 pair of opposite sides is equal and parallel thenthe quadrilateral is parallelogram(opp. sides eq. and //)

Rhombus Definition : a // gram or a kite of 4 equal sides

Properties:

2 pairs of opposite sides are equal.(opp. sides of // gram)

2 pairs of opposite angles are equal (opp. s of // gram)

Diagonals bisect each other(diag. Of // gram)

Diagonals bisects each interior angle

Diagonals are

Rectangle Definition : a parallelogram of 4 right angles

Properties:

2 pairs of opposite sides are equal.(opp. sides of // gram)

2 pairs of opposite angles are equal (opp. s of // gram)

Diagonals bisect each other(diag. Of // gram)

Diagonals are equal

Square Definition : a // gram of 4 right angles and 4 equal sides

Properties:2 pairs of opposite sides are equal.(opp. sides of // gram)

2 pairs of opposite angles are equal (opp. s of // gram)

Diagonals bisect each other(diag. Of // gram)

Diagonals are equal

Diagonals are

450

Angles between each diagonal and each side is 450

Example 1: In the figure, PQRS is a kite

(a) Find x and y.(b) Find the perimeter of the kite PQRS

P

R

SQ

x+1 y+3

8x+y

PQ = PS (given)x+1 = y+3x-y=2 (1)

QR=SR (given)x+y=8 (2)

(1)+(2), 2x=10x=5

Put x=5 into (1), 5-y=2y=3

(a)

(b) PQ = x+1=5+1=6 PQ+PS+SR+QR = 6 + 6 + 8 + 8 =28

Example 2: In the figure, ABCD is a kite. E is a point of intersection of diagonals AC and BD, AE=9 cm, EC=16 cm and DE=EB=12 cm(a) Find the area of ABCD.(b) Find the perimeter of ABCD

(a) ABC= ADC (axis of symmetry AC)AED=900

In ADE,AD2=AE2+DE2=92+122=225 cm2 (Pyth theorem) AD=15 cm

In CDE,DC2=DE2+EC2=122+162=400 cm2 (Pyth theorem) DC=20 cm

Perimeter of ABCD=AD+AB+ DC+CB = 15 + 15 + 20 + 20 =70 cm

A C

D

B

916

12E

12

Area of ADC =

2150

12)169(2

12

1

cm

DEAC

Area of kite ABCD=Area of ABC+Area of ADC = 150+150 =300 cm2

(b)

Example 3: In the figure, ABCD is a parallelogram. Find x and y.

AD//BC (Given)x+680=1800 (prop. Of trapezium) x=1120

(1500-y)+2y=1800 (prop. Of trapezium) 1500+y=1800 y=1800 -1500=300

A

B

D

C

1500-y

680

x

2y

Example 4: In the figure, ABCD is a parallelogram. Find x and y.

DAB=DCB (opp. s of // gram)x+200=3x-100

2x=300

x=150

DAB+CBA=1800 (int.s , AD//BC)x+200+y=1800

150+200+y=1800

y=1450

x+200y

3x+100

A B

CD

Example 5: In the figure, ABCD is a isosceles trapezium with AB=DC.Find x , y and z

1260

xy

z

A

B C

D AD//BC (Given)x+1260=1800 (prop. Of trapezium) x=540

Construct AE // DCE

a

AD//EC and AE//DCADCE is a parallelogram (Definition of // gram)

ADCE is a parallelogram (proof)AE=DC (opp.sides of // gram)

In ABE, AE=DC (proof) AB=AC (given) AB=AE y=a (base s. isos ) a= x (corr. s. AE//DC) y=x =540 y+z=1800 (prop. Of trapesium)z= 1800-540

= 1260

A

NM

B C

MID-POINT THEOREM

IF AM = MB and AN =NC then(a) MN // BC

(b) MN = BC2

1

(Abbreviation: Mid-point theorem)

Example 13: In the figure, ABC is a triangle, find x and y.

DE//AC (mid-point theorem)

x = EDB=420

(corr. s , DE//AC)

BCDE2

1 (mid-point theorem)

y2

16

1262 y

C

E

BDA

y

6

420x

CE=BE (given)AD=DB (given)

Example 14: Prove that BPQR is a parallelgram

AR=RB (given)

(given)

BCRQandBCRQ2

1// (mid-point theorem)

AQ=QC

PCBP (given)

BC2

1

BPRQ

ramparalaisBPQR log(opp-sides eq. And //)

A

Q

CPB

R

Ex 11D1(b)

DA

M

B CN

y cmx cm

5 cm

AM=AC (given)BN=NC (given)

102

15

2

1

x

x

ABMN (mid-point theorem)

BM=MD (given)BN=NC (given)

102

15

2

1

y

y

CDMN(mid-point theorem)

Ex 11D2(b)

AP=BP (given)AQ=CQ (given)

BCPQ // (mid-point theorem)

PQ

BC

A

a1100

460

046 PBCAPQ (corr.s. PQ//BC)

In APQ,APQ+ PAQ+ a = 1800 460+1100+a=1800

a=240

(adj s. on a st line)

A

F E

B D C

10

9

8

3(a)

3(b)

B D C9

A

F E

60

70 50

4.

A Q

P

C

R

B

6

8

(mid-point theorem)

AQ=QB (given)AP=PC (given)

cmBC

BC

BCPQ

162

18

2

1

(mid-point theorem)

BP=PA (given)CR=RB (given)

cmAB

AB

ABPR

122

16

2

1

Area of ABC

296

2

16125

2

cmx

BCAB

INTERCEPT THEOREM

A

B

CD

P

Q

X

Y

transversal

inte

rcep

t

INTERCEPT THEOREM

A B

C D

E F

If AB//CD//EF then

CE

AC

DF

BD (intercept theorem)

INTERCEPT THEOREM

A

C D

E F

CE

AC

DF

AD (intercept theorem)

Construct GB through A such that BG//CD//EF

BGGB//CD//EF (given)

Proved:

CE

AC

DF

AD

Example 15. AP//BQ//CR, AB=BC, AP=11 and CR=5. Find BQ.A

BC

P Q R

5

11 S

Join AR to cut BQ at S

AP//BQ//CR (given)BCAB (given)

1BC

AB

QR

PQ

SR

AS(intercept theorem)

QRPQandSRAS

,ARCInBCAB (given)

SRAS (proved)

5.22

5

2

CRBS (mid-pt theorem)

,APRInSRAS (proved)

QRPQ (proved)

5.52

11

2

APSQ (mid-pt theorem)

BQ=BS+SQ = 2.5+5.5=8

Example 16. AB and DC are straight lined. Find x and y.

Join DE through A and // BC

DE//PQ//BC (given)

QC

AC

PB

AB

(intercept theorem)

A

P Q

B C

(a) Proved: ED

QC

AQ

PB

AP

QC

QCAC

PB

PBAB

11 QC

AC

PB

AB

QC

AC

PB

AB

(b) AB=6, PB=2 and AQ=9. Find QC

QC

AC

PB

AB

QC

QC

9

2

6

QCQC 9392 QC5.4QC

(proved)

Example 16. Find QR and CD.

AP//BQ//CR (given)

QR

PQ

BC

AB

QR

2

6

3

3

62QR

4

(intercept theorem) (intercept theorem)RS

QR

CD

BC

8

46

CD

12

BQ//CR//DS (given)

4

68CD

A P

R

8D S

B Q3 2

6C