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Theorems :)

  • 11-06-2010 06:44PM
    #1
    tracy wacy x
    Closed Accounts Posts: 150 ✭✭


    Theorem one. vertically opposite angles are equal in measure .

    Given : intersecting lines L & K with vertically opposite angles 1 & 3

    To prove : 1 = 3

    Proof : 1 + 2 = 180 ... straight angle
    2 + 3 = 180 ... straight angle
    ~> 1 + 2 = 2 + 3
    then you subtract 2 from each side
    Soo 1 = 3 :)
    Tagged:


«1

Comments

  • #2
    tracy wacy x
    Closed Accounts Posts: 150 ✭✭tracy wacy x


    Theorem 2. the measure of the three angles in a triangle sum up to 180.

    Given : A triangle abc containing angles 1 2 & 3

    To prove : 1 + 2 + 3 = 180

    proof : 4 + 2 + 5 = 180 .. straight angle
    & 1 = 4 and 3 = to 5 because they are alternate
    = 1 + 2 + 3 also equal 180.


  • #3
    tracy wacy x
    Closed Accounts Posts: 150 ✭✭tracy wacy x


    Theorem 3. An exterior angle of a triangle equals the two interior opposite angles in measure.

    Given : triangle abc containing angles 1 2 and 3base is extended to D with exterior angle 4.

    To prove : 4 = 1 + 2

    Proof : 1+ 2 + 3 = 180
    4 +3 = 180 because its a straight line triangle
    ~> 1 + 2 + 3 = 4 + 3
    subtract 3 from each side 1+2 =4


  • #4
    tracy wacy x
    Closed Accounts Posts: 150 ✭✭tracy wacy x



    Theorem 4. if a line passes through a point t on a circle & is perpendicular to the diameter at t, then the line is a tangent to the circle at t.

    Given : A line L, perpendicular to the diameter act, L meets the diameter at point t.

    To prove : That L is a tangent to the circle.

    Proof : [xc] is the longest side of triangle ctx
    ~> [xc] > radius [ct]
    ~> x must lie outside the circle & all the other points lie outside the
    cirle except t.
    L meets the circle at one point only
    L is tangent to the circle.


  • #5
    mista_2k10
    Closed Accounts Posts: 329 ✭✭mista_2k10


    You must have nothing else for doing :rolleyes:... but at least your going to the trouble of writing out the theorems!:);)


  • #6
    lolzNato
    Closed Accounts Posts: 46 ✭lolzNato


    thats cool, but it hurts my eyes :pac: haha


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  • #7
    tracy wacy x
    Closed Accounts Posts: 150 ✭✭tracy wacy x


    just trying to learn them and people are always asking to put them up :rolleyes:


  • #8
    mista_2k10
    Closed Accounts Posts: 329 ✭✭mista_2k10


    tracy wacy x wrote: »
    just trying to learn them and people are always asking to put them up :rolleyes:
    Good idea!;) anyone have any predictions on what theorems will come up?


  • #9
    tracy wacy x
    Closed Accounts Posts: 150 ✭✭tracy wacy x


    Theorem 5. The measure of the angle at the centre of a circle is twice the measure of the angle of circumference , standing on same arc.

    Given : Circle with centre C, containing a , b & d

    To prove : [<bcd] = 2 [<bad]
    Proof : 6= 1 + 2 exterior angle
    but 1 =2 = [ac] = [bc]
    ~> 6 = 2(2)
    5 = 3 + 4 exterior angle
    but 3 =4 [ac] = [cd]
    ~> 5 = 2 (3) [ii]
    add & [ii]
    6 + 5 = 2 (2) + 2 (3)
    ~> 6 + 5 = 2 (2 + 3)
    ~> [<bcd] = 2[<bad]


  • #10
    tracy wacy x
    Closed Accounts Posts: 150 ✭✭tracy wacy x


    Pythagorean theorem :cool:


  • #11
    soccymonster
    Registered Users, Registered Users 2 Posts: 9,553 ✭✭✭soccymonster


    Could you change the font? It burns! (and i'm sure JC'ers would prefer to not strain their retinas :o)


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  • #12
    mista_2k10
    Closed Accounts Posts: 329 ✭✭mista_2k10


    tracy wacy x wrote: »
    Pythagorous theorem :cool:
    :rolleyes:


  • #13
    Luno
    Registered Users, Registered Users 2 Posts: 6,840 ✭✭✭Luno


    mista_2k10 wrote: »
    :rolleyes:

    It can be said both ways...


  • #14
    mista_2k10
    Closed Accounts Posts: 329 ✭✭mista_2k10


    M&S* wrote: »
    It can be said both ways...
    ya learn something new every day!:pac:


  • #15
    Gentlemanne
    Registered Users, Registered Users 2 Posts: 386 ✭✭Gentlemanne


    tracy wacy x wrote: »
    To prove : 1 = 3

    Good luck proving that. :cool:


  • #16
    thebillynator
    Registered Users, Registered Users 2 Posts: 855 ✭✭✭thebillynator


    Gentlemanne wrote: »
    Good luck proving that. :cool:

    should have said
    Angle 1 = Angle 3

    now that can be proven.....


  • #17
    tracy wacy x
    Closed Accounts Posts: 150 ✭✭tracy wacy x


    Deduction 1 . All angles at the circumference on same arc are equal in equal in measure.

    To prove : angle 1 = angle 2

    Proof : point e is in the centre of circle.
    Angle 1 & 3 are both standing on arc [bc].
    3 = 2 [1] from theorem
    [ii] Point E is in centre of circle,
    Angle 2 & angle 3 are both standing on same arc
    ~> 3 = 2 [2]
    therefore 2(1) = 2(2) so
    angle one = two :o


  • #18
    tracy wacy x
    Closed Accounts Posts: 150 ✭✭tracy wacy x


    Deduction 2. An angle subtended by a diameter at the circumference is a right angle

    Given : Circle with centre D. , Point a & c are end point on the diameter , and point . B is on the circle.

    to prove : [<abc] = 90

    Proof : Point . D is the centre of the circle.
    Angle 1 & 2 are both standing on arc ac.

    ~> 2 = 2(1)
    but (2) = 180 straight angle
    and (1) = 90 right angle.


  • #19
    tracy wacy x
    Closed Accounts Posts: 150 ✭✭tracy wacy x


    Deduction 3. the sum of opposite angles of a cyclic quadrilateral is 180.

    To prove : [<abc] + [<adc] = 180

    Proof (i) : Angle 3 is at the centre of the circle . Angles 3 & 1 are both on the arc (ac) ~> 3 = 2 (1)

    (ii) : Angle 3 is at the centre of the circle. Angle @ & $ are both on the arc [ac] ~> 2 = 2[4]
    Add (i) & (ii)
    3 + 2 = 2 [1] + 2 [4] ~> 3 + 2 = 2(1) + (4) but 2 +3 = 360
    ~> 1 + 2 = 180 straight angle

    Hate this one :[


  • #20
    tracy wacy x
    Closed Accounts Posts: 150 ✭✭tracy wacy x


    Theorem 6. If two sides of a triangle are equal in measure then the angles opposite these sides are equal in measure.

    Given : A triangle abc with [ab] equaling [ac] * base angles 1 & 2

    To prove : [<1] = [<2]

    Proof : Consider triangle abd and triangle adc
    (i) [ab] = [ac] ... given
    [ad] = [ad] ... common
    (ii) (o) = (o) [ad] bisects <bac
    (iii) therefore triangle abd is congruent to triangle abc ... SAS
    ~> [<1] = [<2] ~ corresponding angles
    :pac:


  • #21
    tracy wacy x
    Closed Accounts Posts: 150 ✭✭tracy wacy x


    Theorem 7. A line through the centre of a circle perpendicular to a chord bisects the chord.

    Given : Circle with centre C, line M through C , chord [ab] perpendicular to mm
    M n [ab] = d

    To prove : [ad] = [db]

    Proof : Consider triangle adc & triangle bcd
    (i) [dc] = [dc] ... common
    (ii) (x) = (x) both 90
    (iii) the triangle adc is congruent to triangle bcd = RHS
    [ad] = [bd] .... corresponding sides :D



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  • #22
    tracy wacy x
    Closed Accounts Posts: 150 ✭✭tracy wacy x


    do de rest tomoorra :)


  • #23
    niamh x
    Closed Accounts Posts: 8 ✭niamh x


    thanks for saving my life tracy :)


  • #24
    kevin12345
    Registered Users, Registered Users 2 Posts: 1,171 ✭✭✭kevin12345


    I'll stick to learning them from the book...the bold, italic and colours hurt my eyes :pac:

    Thanks anyway :D


  • #25
    niamh x
    Closed Accounts Posts: 8 ✭niamh x


    tracy wacy x wrote: »
    Pythagorean theorem :cool:





    hope so :)


  • #26
    JC Man
    Closed Accounts Posts: 18 ✭JC Man


    What is the difference in deductions and theorems?


  • #27
    Bears and Vodka
    Registered Users, Registered Users 2 Posts: 2,249 ✭✭✭Bears and Vodka


    Your font colour choices is abysmal.
    But thanks for the effort.
    Want to do really well in P2 as I failed P1 miserably.


  • #28
    tracy wacy x
    Closed Accounts Posts: 150 ✭✭tracy wacy x


    is this better :D


  • #29
    WanderingSoul
    Registered Users, Registered Users 2 Posts: 2,419 ✭✭✭WanderingSoul


    tracy wacy x wrote: »
    is this better :D

    Yes! Much better! :D


  • #30
    mista_2k10
    Closed Accounts Posts: 329 ✭✭mista_2k10


    Do we all know our theorems for tomorrow then?:pac:


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  • #31
    kevin12345
    Registered Users, Registered Users 2 Posts: 1,171 ✭✭✭kevin12345


    mista_2k10 wrote: »
    Do we all know our theorems for tomorrow then?:pac:

    Just finished learning the first 6 doing a small bit of CSPE now then finish the theorems off!


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