Space is flat but spacetime is curved? In the picture below a triangle has been drawn on a spherical shaped object. The angles add up to 67+48+73=188 degrees. Since the surface of earth is also almost spherical therefore an experiment can be conducted by three persons who are standing many, many kilometers away from each other. They can form a triangle using long ropes and the angles of such a triangle would sum up more than 180 degrees since earth has positive curvature.

Source: https://www.reddit.com/r/puzzles/comments/89rxsb/using_this_trianglelike_shape_find_the_dimensions/
It is often said that the universe is flat. I think it needs some clarification. The universe in itself is not some solid object. For example, in the video, https://youtu.be/blSTTFS8Uco?t=101, and at many other countless places on internet, they show solid surfaces as if we are supposed to measure the curvature of universe by standing on some solid surface. In simple terms, the universe is made up of matter and space. I would say that by the phrase "universe is flat" what they really mean is that the space is Euclidean or flat. But, then, at the same time it is said that space (or, universe) is Euclidean but spacetime is curved. Well, IMHO, the spacetime, space, and universe go hand in hand. How can you separate spacetime and space from each other?!
The quote below is taken from the following web article https://blogs.scientificamerican.com/degrees-of-freedom/httpblogsscientificamericancomdegrees-of-freedom20110731what-do-you-mean-the-universe-is-flat-part-ii/

Space in Outer Space
So here we come to the basic fact of life that I was referring to at
the beginning of this post. The curvature of space itself.
To avoid any confusion caused by the Earth, take a trip to outer
space. You could think of a  spacecraft tracing a triangle or a square
by traveling in space. That would not be ideal, though, because it
raises all sort of thorny issues about what exactly it means for a
spacecraft to fly straight ahead or to turn by 90 degrees to the left.
Instead, you and two buddies each have a spaceship, and each of the
three travels to some place in the near universe. Once you’re there,
you point lasers at one another and form a triangle of beams.
Now each of you can measure the angle between the two beams that go in
or out of the respective spaceship.
Fact: Those three angles won’t always add up to 180 degrees.
You could do the appropriate calculations and realize that this fact
is a consequence of Einstein’s general theory of relativity. Or you
could distrust math and physics and just go out to space to see for
yourself.
Regardless, this is what it means for space to be curved. Whenever you
can find three points in space, and join them with laser beams, and
find that the triangle doesn’t have the expected 180 degrees, that
means that space is curved.
And when no matter where the spacecraft are the angles add up to 180
degrees--that is what it means for space to be flat.

Does it mean one should conduct such an experiment of creating a triangle using laser beams in an empty space devoid of matter? Since as long as there is matter the General Relativity is going to play its role and light is going bend around and angles will add up to more than 180 degrees. What do they really mean when it is said that universe (or, space) is flat? As a layman, I always get confused. I have checked several sources and it hasn't helped me.
 A: 
What do they really mean when it is said that universe (or, space) is flat?

When they say that space is flat they mean that Euclidean geometry works in space. Meaning that the Pythagorean theorem holds, parallel lines don’t intersect, the sum of angles in a triangle are 180 degrees, etc.
However, what is perhaps more interesting is to consider what is meant by space. As you pointed out, what we have in the universe is curved spacetime. So how can we have curved spacetime and yet flat space? It seems that a distinction between space and spacetime is being drawn.
Consider a solid block of cheese. For all practical purposes the 3D geometry is flat, meaning that Euclid’s axioms hold. Now, you can slice the cheese with a knife and get a series of 2D slices, and Euclid’s axioms would hold on each 2D slice, so the slices would be flat too.
However, it would also be possible to make a curved knife, like a sharp-edged spoon. You could slice the block of cheese into a series of 2D scoop-shaped slices, and Euclid’s axioms would not hold on these slices, so they are not flat.
So the point is that a flat 3D space can be “foliated” (or sliced) into a series of curved 2D sub-spaces. The foliation is not part of the original 3D block, it is arbitrary. And different arbitrary foliations can have different geometric properties.
Now, the universe is a curved 4D spacetime, but cosmologists have a standard way of foliating spacetime into a time-series of space slices. The foliation is arbitrary, but particularly convenient, so it is used ubiquitously in the cosmology community. How curved the slices are in that standard foliation is a parameter in the cosmological model, and when they say space is flat they mean that parameter is zero to within the available observational accuracy.
So basically this is the reverse of the cheese. We have a curved 4D spacetime that can be sliced into flat 3D slices we call space.
A: A good criterion for there being curvature which also works in spacetime is that the parallel transport of a vector around a loop gives you a different vector than the one you started with.
Angles don't work the same way in spacetime as they do in Euclidean space, so even in flat Minkowski spacetime the sum of the interior angles of a spacetime triangle will not equal 180 degrees (whatever that might mean, see below).
Suppose you have a triangle involving one spacelike side of distance $x$, and then two timelike sides which meet at a time $t$ later. The spacelike side has unit 4-vector (0,1). One of the two timelike sides let's take to be at rest so it has unit 4-vector (1,0). The other timelike side must move with velocity $v=x/t$ so it has unit 4-vector $(\gamma, -\gamma v)$.
We ordinarily define an angle between two vectors as the argument of the cosine that you get when you take the dot product. But the dot product between the two timelike unit vectors gives you $$(1,0)\cdot (\gamma, -\gamma v)=\gamma$$ which is always greater than one! So you can't define an angle between timelike vectors in the same way as for Euclidean space. You can say $\gamma=\cosh w$ where $w$ is a rapidity, but that is a different thing.

