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.
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
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
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.
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 straightABC 180
B C
33. Angle Theorems
D Angles in a straight line add up to 180
A
x 30 x 30 180 straightABC 180
B C x 150
34. Angle Theorems
D Angles in a straight line add up to 180
A
x 30 x 30 180 straightABC 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 straightABC 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 straightABC 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 straightABC 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 straightABC 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 straightABC 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 straightABC 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 straightABC 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
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
ae 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
ae corresponding' s , || lines
b f
cg
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
ae corresponding' s , || lines
b f
cg
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
ae corresponding' s , || lines
b f
cg
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
ae corresponding' s , || lines
b f
cg
d h
Cointerior angles (C) are supplementary
c e 180 cointerior' s 180, || lines
d f 180
53. e.g. A B
120
P
y
x 65
C Q D
x 65 180 straightCQD 180
54. e.g. A B
120
P
y
x 65
C Q D
x 65 180 straightCQD 180
x 115
55. e.g. A B
120
X P Y
y
x 65
C Q D
x 65 180 straightCQD 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 straightCQD 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 straightCQD 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 straightCQD 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 straightCQD 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 straightCQD 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 straightCQD 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