11x1 t08 01 angle theorems (2011)
TRANSCRIPT
Euclidean Geometry
Euclidean Geometry Geometry Definitions
Notation
toparallel is ||
Euclidean Geometry Geometry Definitions
Notation
toparallel is ||
lar toperpendicu is
Euclidean Geometry Geometry Definitions
Notation
toparallel is ||
lar toperpendicu is
tocongruent is
Euclidean Geometry Geometry Definitions
Notation
toparallel is ||
lar toperpendicu is
tocongruent is
similar to is |||
Euclidean Geometry Geometry Definitions
Notation
toparallel is ||
lar toperpendicu is
tocongruent is
similar to is |||
therefore
Euclidean Geometry Geometry Definitions
Notation
toparallel is ||
lar toperpendicu is
tocongruent is
similar to is |||
therefore
because
Euclidean Geometry Geometry Definitions
Notation
Terminology
toparallel is ||
lar toperpendicu is
tocongruent is
similar to is |||
therefore
because
produced – the line is extended
Euclidean Geometry Geometry Definitions
Notation
Terminology
toparallel is ||
lar toperpendicu is
tocongruent is
similar to is |||
therefore
because
produced – the line is extended
X Y XY is produced to A
A
Euclidean Geometry Geometry Definitions
Notation
Terminology
toparallel is ||
lar toperpendicu is
tocongruent is
similar to is |||
therefore
because
produced – the line is extended
X Y XY is produced to A
A
YX is produced to B
B
Euclidean Geometry Geometry Definitions
Notation
Terminology
toparallel is ||
lar toperpendicu is
tocongruent is
similar to is |||
therefore
because
produced – the line is extended
X Y XY is produced to A
A
YX is produced to B
B
foot – the base or the bottom
Euclidean Geometry Geometry Definitions
Notation
Terminology
toparallel is ||
lar toperpendicu is
tocongruent is
similar to is |||
therefore
because
produced – the line is extended
X Y XY is produced to A
A
YX is produced to B
B
foot – the base or the bottom
A B
C D
B is the foot of the perpendicular
Euclidean Geometry Geometry Definitions
Notation
Terminology
toparallel is ||
lar toperpendicu is
tocongruent is
similar to is |||
therefore
because
produced – the line is extended
X Y XY is produced to A
A
YX is produced to B
B
foot – the base or the bottom
A B
C D
B is the foot of the perpendicular
collinear – the points lie on the same line
Euclidean Geometry Geometry Definitions
Notation
Terminology
toparallel is ||
lar toperpendicu is
tocongruent is
similar to is |||
therefore
because
produced – the line is extended
X Y XY is produced to A
A
YX is produced to B
B
foot – the base or the bottom
A B
C D
B is the foot of the perpendicular
collinear – the points lie on the same line
concurrent – the lines intersect at the same
point
Euclidean Geometry Geometry Definitions
Notation
Terminology
toparallel is ||
lar toperpendicu is
tocongruent is
similar to is |||
therefore
because
produced – the line is extended
X Y XY is produced to A
A
YX is produced to B
B
foot – the base or the bottom
A B
C D
B is the foot of the perpendicular
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
Naming Conventions
Angles and Polygons are named in cyclic order
Naming Conventions
Angles and Polygons are named in cyclic order
A
B C
CBAABC ˆor
BB ˆor
ambiguity no If
Naming Conventions
Angles and Polygons are named in cyclic order
A
B C
CBAABC ˆor
BB ˆor
ambiguity no If
P
V
T
Q
VPQT or QPVT
or QTVP etc
Naming Conventions
Angles and Polygons are named in cyclic order
A
B C
CBAABC ˆor
BB ˆor
ambiguity no If
P
V
T
Q
VPQT or QPVT
or QTVP etc
Parallel Lines are named in corresponding order
Naming Conventions
Angles and Polygons are named in cyclic order
A
B C
CBAABC ˆor
BB ˆor
ambiguity no If
P
V
T
Q
VPQT or QPVT
or QTVP etc
Parallel Lines are named in corresponding order
A
B
C
D
AB||CD
Naming Conventions
Angles and Polygons are named in cyclic order
A
B C
CBAABC ˆor
BB ˆor
ambiguity no If
P
V
T
Q
VPQT 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 Conventions
Angles and Polygons are named in cyclic order
A
B C
CBAABC ˆor
BB ˆor
ambiguity no If
P
V
T
Q
VPQT 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 = SR
PS = QR
Constructing Proofs
When constructing a proof, any line that you state must be one of the
following;
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.
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.
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), any theorem
you are using must be explicitly stated, whether it is in the algebraic
statement or the reason itself.
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), any theorem
you are using must be explicitly stated, whether it is in the algebraic
statement or the reason itself.
180 sum 18012025 e.g. A
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), any theorem
you are using must be explicitly stated, whether it is in the algebraic
statement or the reason itself.
180 sum 18012025 e.g. A
Your reasoning should be clear and concise
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), any theorem
you are using must be explicitly stated, whether it is in the algebraic
statement or the reason itself.
4. Any working out that follows from lines already stated.
180 sum 18012025 e.g. A
Your reasoning should be clear and concise
Angle Theorems
Angle Theorems 180 toup add linestraight ain Angles
Angle Theorems
A B C
D 180 toup add linestraight ain Angles
x 30
Angle Theorems
A B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCxx 30
Angle Theorems
A B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx
150x
x 30
Angle Theorems
A B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx
150x
x 30
equal are angles opposite Vertically
Angle Theorems
A B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx
150x
x 30
equal are angles opposite Vertically
x3 39
Angle Theorems
A B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx
150x
x 30
equal are angles opposite Vertically
are s'opposite vertically 393xx3 39
Angle Theorems
A B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx
150x
x 30
equal are angles opposite Vertically
are s'opposite vertically 393x
13x
x3 39
Angle Theorems
A B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx
150x
x 30
equal are angles opposite Vertically
are s'opposite vertically 393x
13x
x3 39
360 equalpoint aabout Angles
Angle Theorems
A B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx
150x
x 30
equal are angles opposite Vertically
are s'opposite vertically 393x
13x
x3 39
360 equalpoint aabout Angles
y
15
150120
Angle Theorems
A B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx
150x
x 30
equal are angles opposite Vertically
are s'opposite vertically 393x
13x
x3 39
360 equalpoint aabout Angles
360revolution 36012015015 yy
15
150120
Angle Theorems
A B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx
150x
x 30
equal are angles opposite Vertically
are s'opposite vertically 393x
13x
x3 39
360 equalpoint aabout Angles
360revolution 36012015015 y75y
y
15
150120
Parallel Line Theorems a
b
c d
e f h
g
Parallel Line Theorems a
b
c d
e f h
g
Alternate angles (Z) are equal
Parallel Line Theorems a
b
c d
e f h
g
Alternate angles (Z) are equal
lines||,s'alternate fc
Parallel Line Theorems a
b
c d
e f h
g
Alternate angles (Z) are equal
lines||,s'alternate fc
ed
Parallel Line Theorems a
b
c d
e f h
g
Alternate angles (Z) are equal
lines||,s'alternate fc
ed
Corresponding angles (F) are equal
Parallel Line Theorems a
b
c d
e f h
g
Alternate angles (Z) are equal
lines||,s'alternate fc
ed
Corresponding angles (F) are equal
lines||,s'ingcorrespond ea
Parallel Line Theorems a
b
c d
e f h
g
Alternate angles (Z) are equal
lines||,s'alternate fc
ed
Corresponding angles (F) are equal
lines||,s'ingcorrespond ea
fb
gc
hd
Parallel Line Theorems a
b
c d
e f h
g
Alternate angles (Z) are equal
lines||,s'alternate fc
ed
Corresponding angles (F) are equal
lines||,s'ingcorrespond ea
fb
gc
hd
Cointerior angles (C) are supplementary
Parallel Line Theorems a
b
c d
e f h
g
Alternate angles (Z) are equal
lines||,s'alternate fc
ed
Corresponding angles (F) are equal
lines||,s'ingcorrespond ea
fb
gc
hd
Cointerior angles (C) are supplementary
lines||,180s'cointerior 180 ec
Parallel Line Theorems a
b
c d
e f h
g
Alternate angles (Z) are equal
lines||,s'alternate fc
ed
Corresponding angles (F) are equal
lines||,s'ingcorrespond ea
fb
gc
hd
Cointerior angles (C) are supplementary
lines||,180s'cointerior 180 ec
180 fd
e.g. A B
C D
P
Q
x
y65
120
e.g. A B
C D
P
Q
x
y65
120
180straight 18065 CQDx
e.g. A B
C D
P
Q
x
y65
120
180straight 18065 CQDx
115x
e.g. A B
C D
P
Q
x
y65
120
180straight 18065 CQDx
115x
Construct XY||CD passing through P
X Y
e.g. A B
C D
P
Q
x
y65
120
180straight 18065 CQDx
115x
Construct XY||CD passing through P
X Y
QXYXPQ C||,s'alternate 65
e.g. A B
C D
P
Q
x
y65
120
180straight 18065 CQDx
115x
Construct XY||CD passing through P
X Y
QXYXPQ C||,s'alternate 65
XYABXPB ||,180s'cointerior 180120
e.g. A B
C D
P
Q
x
y65
120
180straight 18065 CQDx
115x
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
Q
x
y65
120
180straight 18065 CQDx
115x
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
Q
x
y65
120
180straight 18065 CQDx
115x
Construct XY||CD passing through P
X Y
QXYXPQ C||,s'alternate 65
XYABXPB ||,180s'cointerior 180120 60XPB
common XPBXPQBPQ
125
6065
y
y
e.g.
Book 2
Exercise 8A;
1cfh, 2bdeh,
3bd, 5bcf,
6bef, 10bd,
11b, 12bc,
13ad, 14, 15
A B
C D
P
Q
x
y65
120
180straight 18065 CQDx
115x
Construct XY||CD passing through P
X Y
QXYXPQ C||,s'alternate 65
XYABXPB ||,180s'cointerior 180120 60XPB
common XPBXPQBPQ
125
6065
y
y