The document defines key concepts in Euclidean geometry, including geometry definitions, notation, naming conventions, and angle theorems. Specifically, it introduces notation for parallel lines, perpendicular lines, congruent lines, and similar lines. It also covers naming angles and polygons in cyclic order, naming parallel lines in corresponding order, and marking equal lines with the same symbol. Finally, it presents theorems such as angles in a straight line summing to 180 degrees and vertically opposite angles being equal.

1. Euclidean Geometry

2. Euclidean Geometry
Geometry Definitions
Notation
||  is parallel to

3. Euclidean Geometry
Geometry Definitions
Notation
||  is parallel to
  is perpendicular to

4. Euclidean Geometry
Geometry Definitions
Notation
||  is parallel to
  is perpendicular to
  is congruent to

5. Euclidean Geometry
Geometry Definitions
Notation
||  is parallel to
  is perpendicular to
  is congruent to
|||  is similar to

6. Euclidean Geometry
Geometry Definitions
Notation
||  is parallel to
  is perpendicular to
  is congruent to
|||  is similar to
  therefore

7. Euclidean Geometry
Geometry Definitions
Notation
||  is parallel to
  is perpendicular to
  is congruent to
|||  is similar to
  therefore
  because

8. Euclidean Geometry
Geometry Definitions Terminology
Notation produced – the line is extended
||  is parallel to
  is perpendicular to
  is congruent to
|||  is similar to
  therefore
  because

9. Euclidean Geometry
Geometry Definitions Terminology
Notation produced – the line is extended
||  is parallel to
  is perpendicular to X
XY is produced to A
Y A
  is congruent to
|||  is similar to
  therefore
  because

10. Euclidean Geometry
Geometry Definitions Terminology
Notation produced – the line is extended
||  is parallel to
  is perpendicular to B X
XY is produced to A
Y A
  is congruent to
YX is produced to B
|||  is similar to
  therefore
  because

11. Euclidean Geometry
Geometry Definitions Terminology
Notation produced – the line is extended
||  is parallel to
  is perpendicular to B X
XY is produced to A
Y A
  is congruent to
YX is produced to B
|||  is similar to
foot – the base or the bottom
  therefore
  because

12. Euclidean Geometry
Geometry Definitions Terminology
Notation produced – the line is extended
||  is parallel to
  is perpendicular to B X
XY is produced to A
Y A
  is congruent to
YX is produced to B
|||  is similar to
foot – the base or the bottom
  therefore A
B B is the foot of the perpendicular
  because
D C

13. Euclidean Geometry
Geometry Definitions Terminology
Notation produced – the line is extended
||  is parallel to
  is perpendicular to B X
XY is produced to A
Y A
  is congruent to
YX is produced to B
|||  is similar to
foot – the base or the bottom
  therefore A
B B is the foot of the perpendicular
  because
D C
collinear – the points lie on the same line

14. Euclidean Geometry
Geometry Definitions Terminology
Notation produced – the line is extended
||  is parallel to
  is perpendicular to B X
XY is produced to A
Y A
  is congruent to
YX is produced to B
|||  is similar to
foot – the base or the bottom
  therefore A
B B is the foot of the perpendicular
  because
D C
collinear – the points lie on the same line
concurrent – the lines intersect at the same
point

15. Euclidean Geometry
Geometry Definitions Terminology
Notation produced – the line is extended
||  is parallel to
  is perpendicular to B X
XY is produced to A
Y A
  is congruent to
YX is produced to B
|||  is similar to
foot – the base or the bottom
  therefore A
B B is the foot of the perpendicular
  because
D C
collinear – the points lie on the same line
concurrent – the lines intersect at the same
point
transversal – a line that cuts two(or more)
other lines

16. Naming Conventions
Angles and Polygons are named in cyclic order

17. Naming Conventions
Angles and Polygons are named in cyclic order
A ABC or ABC ˆ
 If no ambiguity 
B C 
 B or B 
ˆ

 

18. Naming Conventions
Angles and Polygons are named in cyclic order
A ABC or ABC ˆ P
 If no ambiguity  V VPQT or QPVT
B C 
 B or B 
ˆ

 Q or QTVP etc

T

19. Naming Conventions
Angles and Polygons are named in cyclic order
A ABC or ABC ˆ P
 If no ambiguity  V VPQT or QPVT
B C 
 B or B 
ˆ

 Q or QTVP etc

T
Parallel Lines are named in corresponding order

20. Naming Conventions
Angles and Polygons are named in cyclic order
A ABC or ABC ˆ P
 If no ambiguity  V VPQT or QPVT
B C 
 B or B 
ˆ

 Q or QTVP etc

T
Parallel Lines are named in corresponding order
B
D
AB||CD
A
C

21. Naming Conventions
Angles and Polygons are named in cyclic order
A ABC or ABC ˆ P
 If no ambiguity  V VPQT or QPVT
B C 
 B or B 
ˆ

 Q or QTVP etc

T
Parallel Lines are named in corresponding order
B
D
AB||CD
A
C
Equal Lines are marked with the same symbol

22. Naming Conventions
Angles and Polygons are named in cyclic order
A ABC or ABC ˆ P
 If no ambiguity  V VPQT or QPVT
B C 
 B or B 
ˆ

 Q or QTVP etc

T
Parallel Lines are named in corresponding order
B
D
AB||CD
A
C
Equal Lines are marked with the same symbol
Q R
PQ = SR
P PS = QR
S

23. Constructing Proofs
When constructing a proof, any line that you state must be one of the
following;

24. Constructing Proofs
When constructing a proof, any line that you state must be one of the
following;
1. Given information, do not assume information is given, e.g. if you
are told two sides are of a triangle are equal don’t assume it is
isosceles, state that it is isosceles because two sides are equal.

25. Constructing Proofs
When constructing a proof, any line that you state must be one of the
following;
1. Given information, do not assume information is given, e.g. if you
are told two sides are of a triangle are equal don’t assume it is
isosceles, state that it is isosceles because two sides are equal.
2. Construction of new lines, state clearly your construction so that
anyone reading your proof could recreate the construction.

26. Constructing Proofs
When constructing a proof, any line that you state must be one of the
following;
1. Given information, do not assume information is given, e.g. if you
are told two sides are of a triangle are equal don’t assume it is
isosceles, state that it is isosceles because two sides are equal.
2. Construction of new lines, state clearly your construction so that
anyone reading your proof could recreate the construction.
3. A recognised geometrical theorem (or assumption), you must
state clearly the theorem you are using

27. Constructing Proofs
When constructing a proof, any line that you state must be one of the
following;
1. Given information, do not assume information is given, e.g. if you
are told two sides are of a triangle are equal don’t assume it is
isosceles, state that it is isosceles because two sides are equal.
2. Construction of new lines, state clearly your construction so that
anyone reading your proof could recreate the construction.
3. A recognised geometrical theorem (or assumption), you must
state clearly the theorem you are using
e.g. A  25  120  180 sum   180

28. Constructing Proofs
When constructing a proof, any line that you state must be one of the
following;
1. Given information, do not assume information is given, e.g. if you
are told two sides are of a triangle are equal don’t assume it is
isosceles, state that it is isosceles because two sides are equal.
2. Construction of new lines, state clearly your construction so that
anyone reading your proof could recreate the construction.
3. A recognised geometrical theorem (or assumption), you must
state clearly the theorem you are using
e.g. A  25  120  180 sum   180
4. Any working out that follows from lines already stated.

29. Angle Theorems

30. Angle Theorems
Angles in a straight line add up to 180

31. Angle Theorems
D Angles in a straight line add up to 180
x  30
A B C

32. Angle Theorems
D Angles in a straight line add up to 180
A
x  30 x  30  180 straightABC  180 
B C

33. Angle Theorems
D Angles in a straight line add up to 180
A
x  30 x  30  180 straightABC  180 
B C x  150

34. Angle Theorems
D Angles in a straight line add up to 180
A
x  30 x  30  180 straightABC  180 
B C x  150
Vertically opposite angles are equal

35. Angle Theorems
D Angles in a straight line add up to 180
A
x  30 x  30  180 straightABC  180 
B C x  150
Vertically opposite angles are equal
3x  39

36. Angle Theorems
D Angles in a straight line add up to 180
A
x  30 x  30  180 straightABC  180  
B C x  150
Vertically opposite angles are equal
3x  39 3x  39 vertically opposite' s are  

37. Angle Theorems
D Angles in a straight line add up to 180
A
x  30 x  30  180 straightABC  180  
B C x  150
Vertically opposite angles are equal
3x  39 3x  39 vertically opposite' s are  
x  13

38. Angle Theorems
D Angles in a straight line add up to 180
A
x  30 x  30  180 straightABC  180  
B C x  150
Vertically opposite angles are equal
3x  39 3x  39 vertically opposite' s are  
x  13
Angles about a point equal 360

39. Angle Theorems
D Angles in a straight line add up to 180
A
x  30 x  30  180 straightABC  180  
B C x  150
Vertically opposite angles are equal
3x  39 3x  39 vertically opposite' s are  
x  13
150  Angles about a point equal 360

15 
y  120

40. Angle Theorems
D Angles in a straight line add up to 180
A
x  30 x  30  180 straightABC  180  
B C x  150
Vertically opposite angles are equal
3x  39 3x  39 vertically opposite' s are  
x  13
150  Angles about a point equal 360

15
y  120

y  15  150  120  360 revolution  360 

41. Angle Theorems
D Angles in a straight line add up to 180
A
x  30 x  30  180 straightABC  180  
B C x  150
Vertically opposite angles are equal
3x  39 3x  39 vertically opposite' s are  
x  13
150  Angles about a point equal 360

15
y  120

y  15  150  120  360 revolution  360 
y  75

42. Parallel Line Theorems
a bd
c f
e h
g

43. Parallel Line Theorems
Alternate angles (Z) are equal a bd
c f
e h
g

44. Parallel Line Theorems
Alternate angles (Z) are equal a bd
c f
c f alternate' s , || lines e h
g

45. Parallel Line Theorems
Alternate angles (Z) are equal a bd
c f
c f alternate' s , || lines e h
d e g

46. Parallel Line Theorems
Alternate angles (Z) are equal a bd
c f
c f alternate' s , || lines e h
d e g
Corresponding angles (F) are equal

47. Parallel Line Theorems
Alternate angles (Z) are equal a bd
c f
c f alternate' s , || lines e h
d e g
Corresponding angles (F) are equal
ae corresponding' s , || lines

48. Parallel Line Theorems
Alternate angles (Z) are equal a bd
c f
c f alternate' s , || lines e h
d e g
Corresponding angles (F) are equal
ae corresponding' s , || lines
b f
cg
d h

49. Parallel Line Theorems
Alternate angles (Z) are equal a bd
c f
c f alternate' s , || lines e h
d e g
Corresponding angles (F) are equal
ae corresponding' s , || lines
b f
cg
d h
Cointerior angles (C) are supplementary

50. Parallel Line Theorems
Alternate angles (Z) are equal a bd
c f
c f alternate' s , || lines e h
d e g
Corresponding angles (F) are equal
ae corresponding' s , || lines
b f
cg
d h
Cointerior angles (C) are supplementary
c  e  180 cointerior' s  180, || lines

51. Parallel Line Theorems
b
Alternate angles (Z) are equal a d
c f
c f alternate' s , || lines e h
d e g
Corresponding angles (F) are equal
ae corresponding' s , || lines
b f
cg
d h
Cointerior angles (C) are supplementary
c  e  180 cointerior' s  180, || lines
d  f  180

52. e.g. A B
120
P
y
x  65
C Q D

53. e.g. A B
120
P
y
x  65
C Q D
x  65  180 straightCQD  180 

54. e.g. A B
120
P
y
x  65
C Q D
x  65  180 straightCQD  180 
x  115

55. e.g. A B
120
X P Y
y
x  65
C Q D
x  65  180 straightCQD  180 
x  115
Construct XY||CD passing through P

56. e.g. A B
120
X P Y
y
x  65
C Q D
x  65  180 straightCQD  180 
x  115
Construct XY||CD passing through P
XPQ  65 alternate' s , XY || CQ 

57. e.g. A B
120
X P Y
y
x  65
C Q D
x  65  180 straightCQD  180 
x  115
Construct XY||CD passing through P
XPQ  65 alternate' s , XY || CQ 
XPB  120  180 cointerior' s  180 , AB || XY 


58. e.g. A B
120
X P Y
y
x  65
C Q D
x  65  180 straightCQD  180 
x  115
Construct XY||CD passing through P
XPQ  65 alternate' s , XY || CQ 
XPB  120  180 cointerior' s  180 , AB || XY 

XPB  60

59. e.g. A B
120
X P Y
y
x  65
C Q D
x  65  180 straightCQD  180 
x  115
Construct XY||CD passing through P
XPQ  65 alternate' s , XY || CQ 
XPB  120  180 cointerior' s  180 , AB || XY 

XPB  60
BPQ  XPQ  XPB common 

60. e.g. A B
120
X P Y
y
x  65
C Q D
x  65  180 straightCQD  180 
x  115
Construct XY||CD passing through P
XPQ  65 alternate' s , XY || CQ 
XPB  120  180 cointerior' s  180 , AB || XY 

XPB  60
BPQ  XPQ  XPB common 
y  65  60
y  125

61. e.g. A B
120
X P Y
y
x  65
C Q D
x  65  180 straightCQD  180 
x  115
Construct XY||CD passing through P
Book 2
XPQ  65 
alternate' s , XY || CQ  Exercise 8A;
XPB  120  180 cointerior' s  180 , AB || XY 

1cfh, 2bdeh,
3bd, 5bcf,
XPB  60
6bef, 10bd,
BPQ  XPQ  XPB common  11b, 12bc,
y  65  60 13ad, 14, 15
y  125