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,

    13ad, 14, 15

    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