Download Properties of Plane Geometry 13-14(revised)

Page 46
ABBREVIATION USED IN DEDUCTIVE GEOMETRY
A.
Properties of Plane Geometry
No.
Diagram
1
a
b
c
b
2
a
d
A
D
a O b
c
C
3
B
a
A
b
a
A
b
A
A
a  b  c  360
s at a pt.
Two straight lines
AB and CD interest
at point O
a  b and c  d
vert. opp. s
AB // CD
ab
corr. s, AB // CD
a=b
AB // CD
corr. s equal
AB // CD
cd
alt. s, AB // CD
c=d
AB // CD
alt. s equal
AB // CD
e  f  180
int. s, AB // CD
e  f  180
AB // CD
int. s supp.
ABC is a 
a  b  c  180
 sum of 
ABC is a 
c1  a  b
ext.  of 
D
B
d
C
a, b and c are angles
at a point
B
c
5(ii)
adj. s on st. line
D
d
C
a  b  180
B
c
5(i)
Abbreviation
D
4(ii)
C
Conclusion
B
4(i)
C
Given Condition
a and b are adjacent
angles on a straight
line
D
B
A
6(i)
e
f
C
D
B
A
6(ii)
e
f
C
D
A
a
7
c
b
B
C
A
a
8
c1
b
B
C
Page 47
No.
Diagram
Given Condition
Conclusion
Abbreviation
AB = AC
B = C
base s, isos. 
AB = AC and
BD = DC
BAD = CAD
and
AD  BC
prop. of isos. 
AB = AC and
AD  BC
BD = CD
and
BAD= CAD
prop. of isos. 
AB = AC and
BAD = CAD
AD  BC and
BD = CD
prop. of isos. 
B = C
AB = AC
sides opp. equal s
AB = BC = AC
A = B = C = 60o
prop. of equil. 
A = B = C
BC = AC = AB
prop. of equil. 
A
9
C
B
A
10a
B
C
D
A
10b
B
C
D
A
10c
B
C
D
A
11
C
B
A
12
C
B
A
13
C
B
a2
14
a1, a2, a3, … an are
the interior angles of
a n-sided convex
polygon
a3
a1
an
x2
x3
15
x1
xn
The sides of an nsided convex
polygon are
produced in order.
a1  a 2  a3  ...  a n
 n  2   180
x1  x 2  x3  ...  x n  360
 sum of polygon
sum of ext. s of
polygon
Page 48
No.
Diagram
Given Condition
Conclusion
Abbreviation
AB = XY and
AC = XZ and
BC = YZ
ABC  XYZ
SSS
AB = XY and
AC = XZ and
A = X
ABC  XYZ
SAS
AB = XY and
A = X and
B = Y
ABC  XYZ
ASA
AB = XY and
A = X and
C = Z
ABC  XYZ
AAS
AB = XY and
AC = XZ and
C = Z = 90o
ABC  XYZ
RHS
ABC  XYZ
AB = XY and
AC = XZ and
BC = YZ
corr. sides,  s
ABC  XYZ
A = X and
B = Y and
C = Z
corr. s,  s
A
C
16
X
B
Z
Y
A
C
17
X
B
Z
Y
A
C
18
X
B
Z
Y
A
C
19
X
B
Z
Y
A
X
20
B
C
Y
Z
A
B
21
C
X
Y
Z
A
B
22
C
X
Y
Z
Page 49
No.
Diagram
Given Condition
Conclusion
Abbreviation
A = X and
B = Y and
C = Z
ABC ~ XYZ
AAA
AB BC CA


XY YZ ZX
ABC ~ XYZ
3 sides prop.
AB AC
and

XY XZ
A = X
ABC ~ XYZ
ratio of 2 sides,
inc. 
ABC ~ XYZ
AB BC CA


XY YZ ZX
corr. sides, s
ABC ~ XYZ
A = X and
B = Y and
C = Z
corr. s, s
ABC is a 
AB + BC > AC
BC + AC > AB
AB + AC > BC
A
C
23
X
B
Z
Y
A
C
24
X
B
Z
Y
A
C
25
X
B
Z
Y
A
C
26
X
B
Z
Y
A
C
27
X
B
Z
Y
A
28
C
B
A
I is the incentre of
ABC
Z
29
I
Y
C
B
X
A
Z
30
I is the centroid of
ABC
Y
I
B
C
X
A
Z
31
Y
I is the orthcentre of
ABC
I
B
X
C
I is the intersection of the
angle bisectors, i.e.
BAX = BAX
ABY = CBY
BCZ = ACZ
I is the intersection of the
medians, i.e.
AZ = ZB
BX = XC
AY = YC
AI BI CI 2



IX IY IZ 1
I is the intersection of the
altitudes, i.e.
AX  BC
BY  AC
CZ  AB
incentre of 
centroid of 
orthocentre of 
Page 50
No.
Diagram
Given Condition
Conclusion
I is the intersection of the
perpendicular bisectors, i.e.
IX  BC and BX = XC
IY  AC and AY = YC
IZ  AB and AZ = ZB
circumcentre of 
ABCD is a //gram
AB = DC and AD = BC
opp. sides of //gram
ABCD is a //gram
A = C and B = D
opp. s of //gram
ABCD is a //gram
and O is the
intersection of
diagonals
AO = OC and BO = OD
diags. of //gram
AB = DC and
AD = BC
ABCD is a //gram
opp. sides equal
A = C and
B = D
ABCD is a //gram
opp. s equal
AO = OC and
BO = OD
ABCD is a //gram
diags. bisect each
other
AD = BC and
AD // BC
ABCD is a //gram
opp. sides equal
and //
ABCD is a rectangle
All properties of a //gram
ABCD is a rectangle
All the interior angles are
right angles
ABCD is a rectangle
Diagonals are equal
(AC = BD)
A
Z
Y
32
I is the circumcentre
of ABC
I
C
B
X
A
D
33
B
C
A
D
34
B
C
A
35
D
O
B
C
A
D
36
B
C
A
D
37
B
C
A
38
D
O
B
C
A
D
39
B
A
C
D
40
B
C
A
D
41
B
C
A
D
42
B
Abbreviation
C
prop. of rectangle
Page 51
No.
Diagram
A
Given Condition
Conclusion
Abbreviation
ABCD is a rectangle
Diagonals bisect each
other into four equal
parts
(AE = EC = BE = DE)
prop. of rectangle
D
43
E
B
C
A
D
44
ABCD is a square
B
C
A
D
ABCD is a square
45
B
C
A
D
46
C
A
D
All sides are equal
prop. of square
Y
B
All properties of a
rectangle
47
ABCD is a square
Diagonals are
perpendicular to each
other (AC  BD)
ABCD is a square
Angles between each
diagonal and a side is
45o
ABCD is a rhombus
All properties of a
//gram
ABCD is a rhombus
All sides are equal
ABCD is a rhombus
Diagonals are
perpendicular to each
other (AC  BD)
ABCD is a rhombus
Interior angles are
bisected by the
diagonals
(a = b = c = d
and e = f = g = h)
AM = MB and
AN = NC
MN // BC and
1
MN  BC
2
C
B
D
48
A
C
B
D
49
A
C
B
prop. of rhombus
D
50
A
C
B
D
51
A
e a b
f
c d
g
h
C
B
A
M
52
N
C
B
mid-pt. thm.
Page 52
No.
Diagram
A
D
B
53
E
C
F
L1
L2
L3
Given Condition
Conclusion
Abbreviation
L1 // L2 // L3 and
AB = BC
DE = EF
intercept thm.
AM = MB and
MN // BC
AN = NC
intercept thm.
In ABC, ABC = 90
AB2 + BC2 = AC2
Pyth. thm.
In ABC,
AB2 + BC2 = AC2
ABC = 90
converse of Pyth.
thm.
ABCD is an isos.
trapezium
AD // BC, AB = DC,
AE = DF, AC = DB,
BE = FC, AD = EF,
ABC = DCB,
BAD = CDA.
Nil /
prop. of isos.
trapezium
ABCD is a kite
AB = AD, BC = DC,
ABC = ADC,
a1 = a2, c1 = c2,
b1 = d1, b2 = d2,
AC  BD, BO = DO.
Nil /
prop. of a kite
A
54
N
M
C
B
A
55
B
C
A
56
B
C
D
A
57
B
E
F
C
A
a1 a2
B
58
b1
b2
O
c1 c2
C
d1
d2
D