11 x1 t07 01 angle theorems (2013)
TRANSCRIPT
![Page 1: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/1.jpg)
Euclidean Geometry
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Euclidean Geometry Geometry Definitions
Notation
toparallel is ||
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Euclidean Geometry Geometry Definitions
Notation
toparallel is ||
lar toperpendicu is
![Page 4: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/4.jpg)
Euclidean Geometry Geometry Definitions
Notation
toparallel is ||
lar toperpendicu is
tocongruent is
![Page 5: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/5.jpg)
Euclidean Geometry Geometry Definitions
Notation
toparallel is ||
lar toperpendicu is
tocongruent is
similar to is |||
![Page 6: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/6.jpg)
Euclidean Geometry Geometry Definitions
Notation
toparallel is ||
lar toperpendicu is
tocongruent is
similar to is |||
therefore
![Page 7: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/7.jpg)
Euclidean Geometry Geometry Definitions
Notation
toparallel is ||
lar toperpendicu is
tocongruent is
similar to is |||
therefore
because
![Page 8: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/8.jpg)
Euclidean Geometry Geometry Definitions
Notation
Terminology
toparallel is ||
lar toperpendicu is
tocongruent is
similar to is |||
therefore
because
produced – the line is extended
![Page 9: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/9.jpg)
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
![Page 10: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/10.jpg)
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
![Page 11: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/11.jpg)
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
![Page 12: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/12.jpg)
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
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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
![Page 14: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/14.jpg)
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
![Page 15: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/15.jpg)
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
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Naming Conventions
Angles and Polygons are named in cyclic order
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Naming Conventions
Angles and Polygons are named in cyclic order
A
B C
CBAABC ˆor
BB ˆor
ambiguity no If
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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
![Page 19: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/19.jpg)
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
![Page 20: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/20.jpg)
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
![Page 21: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/21.jpg)
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
![Page 22: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/22.jpg)
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
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Constructing Proofs
When constructing a proof, any line that you state must be one of the
following;
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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.
![Page 25: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/25.jpg)
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.
![Page 26: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/26.jpg)
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.
![Page 27: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/27.jpg)
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
![Page 28: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/28.jpg)
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
![Page 29: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/29.jpg)
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
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Angle Theorems
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Angle Theorems 180 toup add linestraight ain Angles
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Angle Theorems
A B C
D 180 toup add linestraight ain Angles
x 30
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Angle Theorems
A B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCxx 30
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Angle Theorems
A B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx
150x
x 30
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Angle Theorems
A B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx
150x
x 30
equal are angles opposite Vertically
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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
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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
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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
![Page 39: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/39.jpg)
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
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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
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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
![Page 42: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/42.jpg)
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
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Parallel Line Theorems a
b
c d
e f h
g
![Page 44: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/44.jpg)
Parallel Line Theorems a
b
c d
e f h
g
Alternate angles (Z) are equal
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Parallel Line Theorems a
b
c d
e f h
g
Alternate angles (Z) are equal
lines||,s'alternate fc
![Page 46: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/46.jpg)
Parallel Line Theorems a
b
c d
e f h
g
Alternate angles (Z) are equal
lines||,s'alternate fc
ed
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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
![Page 48: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/48.jpg)
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
![Page 49: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/49.jpg)
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
![Page 50: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/50.jpg)
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
![Page 51: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/51.jpg)
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
![Page 52: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/52.jpg)
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
![Page 53: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/53.jpg)
e.g. A B
C D
P
Q
x
y65
120
![Page 54: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/54.jpg)
e.g. A B
C D
P
Q
x
y65
120
180straight 18065 CQDx
![Page 55: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/55.jpg)
e.g. A B
C D
P
Q
x
y65
120
180straight 18065 CQDx
115x
![Page 56: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/56.jpg)
e.g. A B
C D
P
Q
x
y65
120
180straight 18065 CQDx
115x
Construct XY||CD passing through P
X Y
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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
![Page 58: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/58.jpg)
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
![Page 59: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/59.jpg)
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
![Page 60: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/60.jpg)
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
![Page 61: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/61.jpg)
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
![Page 62: 11 x1 t07 01 angle theorems (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022052413/559cd39e1a28ab547f8b4808/html5/thumbnails/62.jpg)
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