quadrilateral 1 pair of // opp. sides one of the diagionals is axis of symmetry 2 diagionals are 2...
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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