11 x1 t07 01 angle theorems (2010)
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Euclidean Geometry
Euclidean GeometryGeometry DefinitionsNotation
toparallelis ||
Euclidean GeometryGeometry DefinitionsNotation
toparallelis || lar toperpendicuis
Euclidean GeometryGeometry DefinitionsNotation
toparallelis || lar toperpendicuis tocongruent is
Euclidean GeometryGeometry DefinitionsNotation
toparallelis || lar toperpendicuis tocongruent is
similar tois |||
Euclidean GeometryGeometry DefinitionsNotation
toparallelis || lar toperpendicuis tocongruent is
similar tois |||
therefore
Euclidean GeometryGeometry DefinitionsNotation
toparallelis || lar toperpendicuis tocongruent is
similar tois |||
therefore because
Euclidean GeometryGeometry DefinitionsNotation
Terminology
toparallelis || lar toperpendicuis tocongruent is
similar tois |||
therefore because
produced – the line is extended
Euclidean GeometryGeometry DefinitionsNotation
Terminology
toparallelis || lar toperpendicuis tocongruent is
similar tois |||
therefore because
produced – the line is extended
X YXY is produced to A
A
Euclidean GeometryGeometry DefinitionsNotation
Terminology
toparallelis || lar toperpendicuis tocongruent is
similar tois |||
therefore because
produced – the line is extended
X YXY is produced to A
A
YX is produced to B
B
Euclidean GeometryGeometry DefinitionsNotation
Terminology
toparallelis || lar toperpendicuis tocongruent is
similar tois |||
therefore because
produced – the line is extended
X YXY is produced to A
A
YX is produced to B
B
foot – the base or the bottom
Euclidean GeometryGeometry DefinitionsNotation
Terminology
toparallelis || lar toperpendicuis tocongruent is
similar tois |||
therefore because
produced – the line is extended
X YXY is produced to A
A
YX is produced to B
B
foot – the base or the bottomA
B
CD
B is the foot of the perpendicular
Euclidean GeometryGeometry DefinitionsNotation
Terminology
toparallelis || lar toperpendicuis tocongruent is
similar tois |||
therefore because
produced – the line is extended
X YXY is produced to A
A
YX is produced to B
B
foot – the base or the bottomA
B
CD
B is the foot of the perpendicular
collinear – the points lie on the same line
Euclidean GeometryGeometry DefinitionsNotation
Terminology
toparallelis || lar toperpendicuis tocongruent is
similar tois |||
therefore because
produced – the line is extended
X YXY is produced to A
A
YX is produced to B
B
foot – the base or the bottomA
B
CD
B is the foot of the perpendicular
collinear – the points lie on the same lineconcurrent – the lines intersect at the same
point
Euclidean GeometryGeometry DefinitionsNotation
Terminology
toparallelis || lar toperpendicuis tocongruent is
similar tois |||
therefore because
produced – the line is extended
X YXY is produced to A
A
YX is produced to B
B
foot – the base or the bottomA
B
CD
B is the foot of the perpendicular
collinear – the points lie on the same lineconcurrent – the lines intersect at the same
pointtransversal – a line that cuts two(or more)
other lines
Naming ConventionsAngles and Polygons are named in cyclic order
Naming ConventionsAngles and Polygons are named in cyclic order
A
B C
CBAABC ˆor
BB ˆor
ambiguity no If
Naming ConventionsAngles and Polygons are named in cyclic order
A
B C
CBAABC ˆor
BB ˆor
ambiguity no IfP
V
T
QVPQT or QPVT or QTVP etc
Naming ConventionsAngles and Polygons are named in cyclic order
A
B C
CBAABC ˆor
BB ˆor
ambiguity no IfP
V
T
QVPQT or QPVT or QTVP etc
Parallel Lines are named in corresponding order
Naming ConventionsAngles and Polygons are named in cyclic order
A
B C
CBAABC ˆor
BB ˆor
ambiguity no IfP
V
T
QVPQT or QPVT or QTVP etc
Parallel Lines are named in corresponding order
A
B
C
D
AB||CD
Naming ConventionsAngles and Polygons are named in cyclic order
A
B C
CBAABC ˆor
BB ˆor
ambiguity no IfP
V
T
QVPQT or QPVT or QTVP etc
Parallel Lines are named in corresponding order
A
B
C
D
AB||CD
Equal Lines are marked with the same symbol
Naming ConventionsAngles and Polygons are named in cyclic order
A
B C
CBAABC ˆor
BB ˆor
ambiguity no IfP
V
T
QVPQT or QPVT or QTVP etc
Parallel Lines are named in corresponding order
A
B
C
D
AB||CD
Equal Lines are marked with the same symbol
P
Q R
S
PQ = SRPS = QR
Constructing ProofsWhen constructing a proof, any line that you state must be one of the following;
Constructing ProofsWhen 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.
Constructing ProofsWhen 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.
Constructing ProofsWhen 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
Constructing ProofsWhen 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
180sum 18012025e.g. A
Constructing ProofsWhen 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
4. Any working out that follows from lines already stated.
180sum 18012025e.g. A
Angle Theorems
Angle Theorems180 toup add linestraight ain Angles
Angle TheoremsA B C
D 180 toup add linestraight ain Anglesx 30
Angle TheoremsA B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCxx 30
Angle TheoremsA B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx150x
x 30
Angle TheoremsA B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx150x
x 30
equalareanglesopposite Vertically
Angle TheoremsA B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx150x
x 30
equalareanglesopposite Vertically
x3 39
Angle TheoremsA B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx150x
x 30
equalareanglesopposite Vertically
ares'oppositevertically 393xx3 39
Angle TheoremsA B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx150x
x 30
equalareanglesopposite Vertically
ares'oppositevertically 393x13x
x3 39
Angle TheoremsA B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx150x
x 30
equalareanglesopposite Vertically
ares'oppositevertically 393x13x
x3 39
360 equalpoint aabout Angles
Angle TheoremsA B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx150x
x 30
equalareanglesopposite Vertically
ares'oppositevertically 393x13x
x3 39
360 equalpoint aabout Anglesy
15150
120
Angle TheoremsA B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx150x
x 30
equalareanglesopposite Vertically
ares'oppositevertically 393x13x
x3 39
360 equalpoint aabout Angles
360revolution 36012015015 yy15
150120
Angle TheoremsA B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx150x
x 30
equalareanglesopposite Vertically
ares'oppositevertically 393x13x
x3 39
360 equalpoint aabout Angles
360revolution 36012015015 y75y
y15
150120
Parallel Line Theoremsa b
c d
efh
g
Parallel Line Theoremsa b
c d
efh
g
Alternate angles (Z) are equal
Parallel Line Theoremsa b
c d
efh
g
Alternate angles (Z) are equal
lines||,s'alternate fc
Parallel Line Theoremsa b
c d
efh
g
Alternate angles (Z) are equal
lines||,s'alternate fced
Parallel Line Theoremsa b
c d
efh
g
Alternate angles (Z) are equal
lines||,s'alternate fced
Corresponding angles (F) are equal
Parallel Line Theoremsa b
c d
efh
g
Alternate angles (Z) are equal
lines||,s'alternate fced
Corresponding angles (F) are equal
lines||,s'ingcorrespond ea
Parallel Line Theoremsa b
c d
efh
g
Alternate angles (Z) are equal
lines||,s'alternate fced
Corresponding angles (F) are equal
lines||,s'ingcorrespond eafb gc hd
Parallel Line Theoremsa b
c d
efh
g
Alternate angles (Z) are equal
lines||,s'alternate fced
Corresponding angles (F) are equal
lines||,s'ingcorrespond eafb gc hd
Cointerior angles (C) are supplementary
Parallel Line Theoremsa b
c d
efh
g
Alternate angles (Z) are equal
lines||,s'alternate fced
Corresponding angles (F) are equal
lines||,s'ingcorrespond eafb gc hd
Cointerior angles (C) are supplementary
lines||,180s'cointerior 180 ec
Parallel Line Theoremsa b
c d
efh
g
Alternate angles (Z) are equal
lines||,s'alternate fced
Corresponding angles (F) are equal
lines||,s'ingcorrespond eafb gc hd
Cointerior angles (C) are supplementary
lines||,180s'cointerior 180 ec180 fd
e.g. A B
C D
P
Qx
y65
120
e.g. A B
C D
P
Qx
y65
120
180straight 18065 CQDx
e.g. A B
C D
P
Qx
y65
120
180straight 18065 CQDx115x
e.g. A B
C D
P
Qx
y65
120
180straight 18065 CQDx115x
Construct XY||CD passing through P
X Y
e.g. A B
C D
P
Qx
y65
120
180straight 18065 CQDx115x
Construct XY||CD passing through P
X Y
QXYXPQ C||,s'alternate 65
e.g. A B
C D
P
Qx
y65
120
180straight 18065 CQDx115x
Construct XY||CD passing through P
X Y
QXYXPQ C||,s'alternate 65
XYABXPB ||,180s'cointerior 180120
e.g. A B
C D
P
Qx
y65
120
180straight 18065 CQDx115x
Construct XY||CD passing through P
X Y
QXYXPQ C||,s'alternate 65
XYABXPB ||,180s'cointerior 180120 60XPB
e.g. A B
C D
P
Qx
y65
120
180straight 18065 CQDx115x
Construct XY||CD passing through P
X Y
QXYXPQ C||,s'alternate 65
XYABXPB ||,180s'cointerior 180120 60XPB
common XPBXPQBPQ
e.g. A B
C D
P
Qx
y65
120
180straight 18065 CQDx115x
Construct XY||CD passing through P
X Y
QXYXPQ C||,s'alternate 65
XYABXPB ||,180s'cointerior 180120 60XPB
common XPBXPQBPQ
1256065
yy
e.g.
Book 2Exercise 8A;1cfh, 2bdeh,3bd, 5bcf,6bef, 10bd,11b, 12bc,
13ad, 14, 15
A B
C D
P
Qx
y65
120
180straight 18065 CQDx115x
Construct XY||CD passing through P
X Y
QXYXPQ C||,s'alternate 65
XYABXPB ||,180s'cointerior 180120 60XPB
common XPBXPQBPQ
1256065
yy
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