# 11 x1 t07 01 angle theorems (2013)

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• 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 or

BB or

ambiguity no If

• Naming Conventions

Angles and Polygons are named in cyclic order

A

B C

CBAABC or

BB or

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 or

BB or

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 or

BB or

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 or

BB or

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 or

BB or

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 dont 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 dont 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 dont 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 dont 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 dont 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 dont 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 ABCx150x

x 30

• Angle Theorems

A B C

D 180 toup add linestraight ain Angles

180straight 18030 ABCx150x

x 30

equal are angles opposite Vertically

• Angle Theorems

A B C

D 180 toup add linestraight ain Angles

180straight 18030 ABCx150x

x 30

equal are angles opposite Vertically

x3 39

• Angle Theorems

A B C

D 180 toup add linestraight ain Angles

180straight 18030 ABCx150x

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 ABCx150x

x 30

equal are angles opposite Vertically

are s'opposite vertically 393x13x

x3 39

• Angle Theorems

A B C

D 180 toup add linestraight ain Angles

180straight 18030 ABCx150x

x 30

equal are angles opposite Vertically

are s'opposite vertically 393x13x

x3 39

360 equalpoint aabout Angles

• Angle Theorems

A B C

D 180 toup add linestraight ain Angles

180straight 18030 ABCx150x

x 30

equal are angles opposite Vertically

are s'opposite vertically 393x13x

x3 39

360 equalpoint aabout Angles

y

15

150120

• Angle Theorems

A B C

D 180 toup add linestraight ain Angles

180straight 18030 ABCx150x

x 30

equal are angles opposite Vertically

are s'opposite vertically 393x13x

x3 39

360 equalpoint aabout Angles

360revolution 36012015015 yy15

150120

• Angle Theorems

A B C

D 180 toup add linestraight ain Angles

180straight 18030 ABCx150x

x 30

equal are angles opposite Vertically

are s'opposite vertically 393x13x

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 fced

• Parallel Line Theorems a

b

c d

e f h

g

Alternate angles (Z) are equal

lines||,s'alternate fced

Corresponding angles (F) are equal

• Parallel Line Theorems a

b

c d

e f h

g

Alternate angles (Z) are equal

lines||,s'alternate fced

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 fced

Corresponding angles (F) are equal

lines||,s'ingcorrespond eafb

gc

hd

• Parallel Line Theorems a

b

c d

e f h

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 Theorems a

b

c d

e f h

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 Theorems a

b

c d

e f h

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

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 CQDx115x

• e.g. A B

C D

P

Q

x

y65

120

180straight 18065 CQDx115x

Construct XY||CD passing through P

X Y

• e.g. A B

C D

P

Q

x

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

Q

x

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

Q

x

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

Q

x

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

Q

x

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

125

6065

y

y

• e.g.

Book 2

Exercise 8A;

1cfh, 2bdeh,

3bd, 5bcf,

6bef, 10bd,

11b, 12bc,

A B

C D

P

Q

x

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

125

6065

y

y