Theorems :)

tracy wacy x

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

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.

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

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.

mista_2k10

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

lolzNato

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

tracy wacy x

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

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?

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]

tracy wacy x

Pythagorean theorem :cool:

soccymonster

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

mista_2k10

tracy wacy x wrote: »

Pythagorous theorem :cool:

:rolleyes:

Luno

mista_2k10 wrote: »

:rolleyes:

It can be said both ways...

mista_2k10

M&S* wrote: »

It can be said both ways...

ya learn something new every day!:pac:

Gentlemanne

tracy wacy x wrote: »

To prove : 1 = 3

Good luck proving that. :cool:

thebillynator

Gentlemanne wrote: »

Good luck proving that. :cool:

should have said
Angle 1 = Angle 3

now that can be proven.....

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

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.

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 :[

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:

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

tracy wacy x

do de rest tomoorra

niamh x

thanks for saving my life tracy

kevin12345

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

Thanks anyway

niamh x

tracy wacy x wrote: »

Pythagorean theorem :cool:

hope so

JC Man

What is the difference in deductions and theorems?

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.

tracy wacy x

is this better

WanderingSoul

tracy wacy x wrote: »

is this better

Yes! Much better!

mista_2k10

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

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!

DylanS09

I tried to learn some a few minutes ago. I failed miserably!