Proofs

I can't offer a full proof, but I can suggest some possibilities that will lead you to your proof. In a parallelogram, you can easily demonstrate that the angles formed by a cord extending between parallel lines and the parallel lines themselves, and that are formed on opposite sides of the cord, are equal. This will work for both pairs of triangles in the parallelogram, and can be applied to all of the angles at the corners of the parallelogram. This will lead you to demonstrating that the pairs of triangles "pointing" to each other (not adjacent pairs) are similar, and in fact congruent. From there it is not difficult to establish that the connected sections of the two interior cords are equal.

An axiom is a statement that is accepted without proof. Proofs are based on statements that are already established, so therefore without axioms we would have no starting point.

Let a be any number and n any positive integer, an/an = 1 since since anything divided by itself is 1. But the laws of exponents say that an/an =an-n=a0 So this proves that a0 =1

DeMorgan's Theorems are:

1. ~(A&B) ~A&~B.

For example:

"It's not raining and cold" means the same as "It is either not raining, or it is not cold", and

"It's not either raining or cold" means the same as "It's not raining and it's not cold".

Applied recursively, DeMorgan's theorem means that to find the inverse of a complex sentence, change all atoms to their inverses, and change all ANDs to ORs and ORs to ANDs.

Any negative integer can be factored to -1 times its positive value. Because negative one times itself is positive one, when multiplied by each other they cancel out.

So if you're multiplying a negative integer A by a negative integer B. Replace A and B with -1*|A| and -1*|B| (You can do this because you know A and B are negative), and use the distributive property to rearrange them. Now you can see the -1*-1 term and equate it to 1, leaving only the |A| and |B| behind. Because two positive numbers multiplied together are always positive, the result will always be positive.

Represented algebraically, as long as A and B are negative integers, the following is true:

AB = -1|A|*-1|B| = -1*-1|AB| = |AB|.

You have to put them over a common denominator.

These numbers can be expressed as:

66/90 35/90 54/90 50/90

So in order they are:

35/90 50/90 54/90 66/90

Or back in their original terms:

7/18 9/9 3/5 11/15

[Alternatively you could convert them all to decimals.]

Theorem: A Proven Statement.

Postulate: An Accepted Statement without Proof.

They mean similar things. A postulate is an unproven statement that is considered to be true; however a theorem is simply a statement that may be true or false, but only considered to be true if it has been proven.

Given:In a triangle ABC in which EF BC

To prove that:AE/EB=AF/FC

Construction:Draw EX perpendicular AC and FY perpendicular AB

Proof:taking the ratios of area of triangle AEF and EBF and second pair of ratio of area of triangle AEF and ECF.

We get AE/EB and AF/FC

we know that triangle lie b/w sme and same base is equal in area

therefore area of EBF I equal to area of ECF

therefore AE/EB=AF/FC

HENCE PROVED

That the sides are equal in length and the interior angles are the same sizes

let x= square root 6 + square root 30, I will write S(6) for square root 6, or S(n) for square root n x-S(6)=S(30) now square both sides x^2-2S(6)+6=30 or x^2-24=2S(6) Now square both sides again to get rid of the radical sign x^4-48x+24^2=24 or after we simplify x^4-48x-552=0 Now we know x is a solution of this polynomial because we created the polynomial to make it so! But the rational roots theorem tells us that all the solutions can be written as p/q where p is an integer factor of 552 and q is an integer factor of 1 ( it is 1or -1) So the solution must be a factor of 552 if the solution is rational. ( there are irrational and complex solutions) 2^3x3x23=552 so the only possible rational roots are combinations of these factors, but we know square root of 5 +square root of 30 is <8, we could say <10 and also by approximation. So that means we only have to deal with the factors 2,3,4 and 6 (all the other combinations are >or= 8) But we also know sqrt6+sqrt(30)>2,3 or 4 because sqrt 4+sqrt 4=4 and our sum is greater. So we are left with 6 Well we know our sum can be written as Sqrt(6)[(Sqrt5)+ 1)] which tells us it is not 6. So it is none of the possible rational roots and hence not rational. Lots of ways to do this last part, just show that it can't be any of the rational factors. following the above notation, If S(6) + S(30) were rational, then it's square would be rational as well so we will show that 6 + 2S(180) + 30 is not rational it suffices to show that S(180) is irrational becasue addition is closed in the rational numbers. 180 = 36 times 5 so S(180) = 6S(5). Now we simply need to show that S(5) is irrational. Assume that (a/b)^2 = 5 where a/b is a fraction in Lowest Terms. then a^2 = 5b^2 so a^2 is divisible by 5 which means that a is divisible by 5 (see footnote). for some number k, a=5k. now we have 25k^2=5b^2 dividing through by 5 we see that b is also divisible by five, contradicting the fact that a/b was in lowest terms. Q.E.D. *every integer has a unique prime facotization. If the prime factorization of a is P1 * P2 * P3 * ... * Pn and 5 is not included, then squaring a simply squares all the prime factors, and 5, wich is prime will still not be part of the prime factorization.

the fraction 22/7 is often used as an approximation for the value of pi. it is off from the actual value by about 0.04 percent (1 in 2500).

This would be like measuring a football field and being off by about 1.5 inches on your measurement.

The fraction 355/113 is amazingly close, better than 1 part in 10 million

2 pi r times the circumference sqaured. then take this and find out its square root.

This however, only works if you have the circumference.

If you have both the volume and the height, you can find the formula for the radius by solving the following literal equation for "r":

V=1/3 r^2(3.14)(H)

r=(3V/pi H)square root

Take the length of the rectangle and multiply it by the width.

For example, if I had a rectangle with two sides 4 inches in length and two sides 3 inches in length, then the area would be 4x3, which equals 12 square inches.

Let G be a complete graph with n vertices. Consider the case where n=2. With only 2 vertices it is clear that there will only be one edge. Now add one more vertex to get n = 3. We must now add edges between the two old vertices and the new one for a total of 3 vertices. We see that adding a vertex to a graph with n vertices gives us n more edges. We get the following sequence

Edges on a graph with n vertices: 0+1+2+3+4+5+...+n-1. Adding this to itself and dividing by two yields the following formula for the number of edges on a complete graph with n vertices: n(n-1)/2.

I'm assuming that you mean 'square root'. Yes, this sum is irrational. So are each of the two numbers alone. A simple proof can be done by writing x=square root 2 + square root 3 and then "squareing away" the square roots and then use the rational roots theorem. The sum or difference of two irrational number need not be irrational! Look at sqrt(2)- sqrt(2)=0 which is rational.

WHOEVER ANSWERED THIS PREVIOUSLY HAS NO IDEA WHAT WE ARE ALL TALKIN ABOUT , OBVIOUSLY

* * * * *

Ok, here is someone who might know.

If the first number is bigger than the second, the difference is positive;

If the first number is the same as the second, the difference is zero; and

If the first number is smaller than the second, the difference is negative.

No. Consider the statement "If I'm alive, then I'm not dead." That statement is true. The converse is "If I'm not dead, then I'm alive.", which is also true.

It's always a common multiple; it's not always least.

Simple counter example:

4 × 6 = 24

But LCM(4, 6) = 12

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Note: HCF(4, 6) = 2

What is true of any two whole numbers that the product of the two numbers is equal to the product of their highest common factor and lowest common multiple.

eg 4 × 6 = hcf(4, 6) × lcm(4, 6) = 2 × 12 = 24.