Learn Basic Postulates in Geometry with Our Experts

If two parallel lines are cut by a transversal, then:
each pair of alternate angles are equal
each pair of alternate angles add up to 180 degrees
each pair of corresponding angles add up to 180 degees.
each pair of corresponding angles is equal.

The opposite faces of a dice always have a total of ___ on them.

What is the greatest number of acute angles that a triangle can contain?

When two lines cross each other at a point we call them ___ lines.

Can you draw a line of length 2.25 centimetres using a ruler? How about using ruler and compass?

Is 90 a perfect square?

Help find out сan a triangle have two obtuse angle?

Parallel lines have no solution. True or false

A line which intersects two or more lines on the same plane is called a
parallel
perpendicular
transversal

A __ is a flat surface in geometry that never ends in two dimensions and has no thickness. A point, a line, a plane, a circle

Logarithms and ratios.
This is the question:
${\log }_b<!--\; \; -->64=\frac{3}{b}$
And have to find b.
So I tried a bit and got this:
$\frac{b}{\log <!--\; \; -->b}=\frac{\log <!--\; \; -->64}{3}$
But have no idea what to do next.
Thanks for your help.

What is the slope of a line parallel to the $y$-axis?

When reading about the history of Euclid's Elements, one finds a pretty length story about the Greeks and Arabs spending countless hours trying to prove Euclid's 5th Postulate.
But I've yet to come across a source stating that "this is the man who finally proved the 5th postulate!"
Has it ever been formally proven, or am I misunderstanding the issue?

Is there a natural number between 0 and 1?
A proof, s'il vous plaît, not your personal opinion. (Assume the Peano Postulates.)

The necessary and sufficient condition for a non - empty subset W of a vector space V(F) to be a subspace of V is
a,b in F and $\mathrm{\alpha <!--\; \alpha \; -->}$, $\mathrm{\beta <!--\; \beta \; -->}$ in W implies a$\mathrm{\alpha <!--\; \alpha \; -->}$ + b$\mathrm{\beta <!--\; \beta \; -->}$ in W
I need to prove the postulates of vector space with this condition. Hints ?

My graph theory book postulates the if a simple graph with n vertices has at least C(n - 1, 2) + 2 edges then the graph must be Hamiltonian.
This is probably true but I am confused by the notation of what C(n - 1, 2) means?
C usually represents a cycle but clearly not in this case. And whatever function they are referencing takes 2 parameters which is quite strange.

Bertrand's postulate says:
For every $n>1$ there is always at least one prime $p$ such that $n<p<2n$.
Is the following statement:
For every $n>3$ there is always at least one prime $p$ such that $F_n<p<F_{n+1}$ ($F_n$ is $n$-th Fibonacci number).
also valid?
If it is invalid, is there a finite or infinite number of ns such that there is no prime between $F_n$ and $F_{n+1}$?

This question is inspiblack by another question. I feel intuitively that it may be interesting, but don't have enough number theory background to tackle it.

Is there a smallest real number $a$ such that there exist a natural number $N$ so that:
$n>N⟹<!--\; ⟹\; -->p_{n+1}\le <!--\; \le \; -->a\cdot <!--\; \cdot \; -->p_n$?
I believe it can be proved that $n>7⟹<!--\; ⟹\; -->p_{n+1}\le <!--\; \le \; -->\sqrt{2}\cdot <!--\; \cdot \; -->p_n$.

Two questions came to mind when I was reading the proof for Bertrand's Postulate (there's always a prime between $n$ and $2n$):
(1) Can we change the proof somehow to show that: $\mathrm{\forall <!--\; \forall \; -->}x>x_0$, there exists a prime $p$$\in <!--\; \in \; -->[x,ax]$, for some $a\in <!--\; \in \; -->(1,2)$?
(2) Suppose the (1) is true, what is the smallest value of $x_0$?
I'm not sure how to prove either of them, any input would be greatly appreciated! And correct me if any of the above statement is wrong. Thank you!

Show that the proposition P:

There exists a pair of straight lines that are at constant distance from each other.

is equivalent to the Parallel Postulate Q :

If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

I tried to prove $Q\to <!--\; \to \; -->P$ then $\mathrm{\neg <!--\; \neg \; -->}Q\to <!--\; \to \; -->\mathrm{\neg <!--\; \neg \; -->}P$. But for the second part, I can do nothing because as soon as the postulate is supposed to be untrue, the equivalent relation between angles no more exist, therefore it's hard to get congruent triangles as I used to do.