11 X1 T06 01 Angle Theorems
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Transcript of 11 X1 T06 01 Angle Theorems
![Page 1: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/1.jpg)
Euclidean Geometry
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Euclidean GeometryGeometry DefinitionsNotation
toparallelis ||
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Euclidean GeometryGeometry DefinitionsNotation
toparallelis || lar toperpendicuis
![Page 4: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/4.jpg)
Euclidean GeometryGeometry DefinitionsNotation
toparallelis || lar toperpendicuis tocongruent is
![Page 5: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/5.jpg)
Euclidean GeometryGeometry DefinitionsNotation
toparallelis || lar toperpendicuis tocongruent is
similar tois |||
![Page 6: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/6.jpg)
Euclidean GeometryGeometry DefinitionsNotation
toparallelis || lar toperpendicuis tocongruent is
similar tois |||
therefore
![Page 7: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/7.jpg)
Euclidean GeometryGeometry DefinitionsNotation
toparallelis || lar toperpendicuis tocongruent is
similar tois |||
therefore because
![Page 8: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/8.jpg)
Euclidean GeometryGeometry DefinitionsNotation
Terminology
toparallelis || lar toperpendicuis tocongruent is
similar tois |||
therefore because
produced – the line is extended
![Page 9: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/9.jpg)
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
![Page 10: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/10.jpg)
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
![Page 11: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/11.jpg)
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
![Page 12: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/12.jpg)
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
![Page 13: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/13.jpg)
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
![Page 14: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/14.jpg)
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
![Page 15: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/15.jpg)
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
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Naming ConventionsAngles and Polygons are named in cyclic order
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Naming ConventionsAngles and Polygons are named in cyclic order
A
B C
CBAABC ˆor
BB ˆor
ambiguity no If
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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
![Page 19: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/19.jpg)
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
![Page 20: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/20.jpg)
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
![Page 21: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/21.jpg)
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
![Page 22: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/22.jpg)
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
![Page 23: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/23.jpg)
Constructing ProofsWhen constructing a proof, any line that you state must be one of the following;
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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.
![Page 25: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/25.jpg)
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.
![Page 26: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/26.jpg)
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
![Page 27: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/27.jpg)
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
![Page 28: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/28.jpg)
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
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Angle Theorems
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Angle Theorems180 toup add linestraight ain Angles
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Angle TheoremsA B C
D 180 toup add linestraight ain Anglesx 30
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Angle TheoremsA B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCxx 30
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Angle TheoremsA B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx150x
x 30
![Page 34: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/34.jpg)
Angle TheoremsA B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx150x
x 30
equalareanglesopposite Vertically
![Page 35: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/35.jpg)
Angle TheoremsA B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx150x
x 30
equalareanglesopposite Vertically
x3 39
![Page 36: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/36.jpg)
Angle TheoremsA B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx150x
x 30
equalareanglesopposite Vertically
ares'oppositevertically 393xx3 39
![Page 37: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/37.jpg)
Angle TheoremsA B C
D 180 toup add linestraight ain Angles
180straight 18030 ABCx150x
x 30
equalareanglesopposite Vertically
ares'oppositevertically 393x13x
x3 39
![Page 38: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/38.jpg)
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
![Page 39: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/39.jpg)
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
![Page 40: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/40.jpg)
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
![Page 41: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/41.jpg)
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
![Page 42: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/42.jpg)
Parallel Line Theoremsa b
c d
efh
g
![Page 43: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/43.jpg)
Parallel Line Theoremsa b
c d
efh
g
Alternate angles (Z) are equal
![Page 44: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/44.jpg)
Parallel Line Theoremsa b
c d
efh
g
Alternate angles (Z) are equal
lines||,s'alternate fc
![Page 45: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/45.jpg)
Parallel Line Theoremsa b
c d
efh
g
Alternate angles (Z) are equal
lines||,s'alternate fced
![Page 46: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/46.jpg)
Parallel Line Theoremsa b
c d
efh
g
Alternate angles (Z) are equal
lines||,s'alternate fced
Corresponding angles (F) are equal
![Page 47: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/47.jpg)
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
![Page 48: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/48.jpg)
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
![Page 49: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/49.jpg)
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
![Page 50: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/50.jpg)
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
![Page 51: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/51.jpg)
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
![Page 52: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/52.jpg)
e.g. A B
C D
P
Qx
y65
120
![Page 53: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/53.jpg)
e.g. A B
C D
P
Qx
y65
120
180straight 18065 CQDx
![Page 54: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/54.jpg)
e.g. A B
C D
P
Qx
y65
120
180straight 18065 CQDx115x
![Page 55: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/55.jpg)
e.g. A B
C D
P
Qx
y65
120
180straight 18065 CQDx115x
Construct XY||CD passing through P
X Y
![Page 56: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/56.jpg)
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
![Page 57: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/57.jpg)
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
![Page 58: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/58.jpg)
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
![Page 59: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/59.jpg)
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
![Page 60: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/60.jpg)
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
![Page 61: 11 X1 T06 01 Angle Theorems](https://reader034.fdocuments.in/reader034/viewer/2022052413/5597df711a28ab5e388b4790/html5/thumbnails/61.jpg)
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