This is a good question with many possible positive answers.    In addition to the incisive observations already given by @Jose Antonio Vargas-Guzmán, we can tackle this question in other ways.

One way to view the goodness and importance of the Euclid of Alexandria is to consider the history of proof.    A very good overview of this aspect of the history of mathematics is given in

S.G. Krantz, The history and concept of mathematical proof,  2007:

http://www.math.wustl.edu/~sk/eolss.pdf

It is observed (p. 4) that Thales (640 B.C.E.-546 B.C.E.), Eudoxus (408 B.C.E.-355 B.C.E.) and Theaetetus of Athens (417 B.C.E.-369 B.C.E.) formulated theorems.   It was Thales who definitely proved some theorems in geometry (later put into a broader context by Euclid).   Eudoxus failed to find means to prove his theorems.

It was Euclid of Alexandria who introduced a framework for doing mathematics, starting with definitions and axioms followed by theorems.   It is observed (p. 5)  that Euclid introduced the paradigm for reasoning and proving in mathematics that has endured for over 20 centuries into the present.   It was Artistotle (384 B.C.E.-322 B.C.E.) who required that a definition must set forth (describe, explain) a concept in terms of known concepts and undefined terms such as set in topology.

Axioms (called Postulates by Euclid) are statements of fact (crisp mathematical assertions) that use defined terms about something that is self-evident.    The penultimate example is Euclid's 5th Parallel Postulate:

Euclidi's  P5.   That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitey, meet on that side on which are the angles less than two right angles.  

[Otherwise the two straight lines are parallel.]   This is from Sir T.L. Heath's translation of Euclid's Elements.   S. Krantz (p. 14) states the 5th Postulate in another way:

[Krantz] P5.  For each line L and each point P that does not lie on L, there is a unique line L' through P such that L' is parallel to L (see attached image).

Ptolemy and Proclus attempted to prove the 5th Postulate.   It was Gerolamo Saccheri (1667-1733), University of Pavia, who first suggested the possibility of postulates other than those in Euclid.    Saccheri rewrote the 5th Postulate in terms of a cluster of lines issuing from a point. Then J.H. Lambert (1728-1777) who published his monograph Theory of Parallels (1766).   Lambert focused on spherical triangles instead of plane triangles because the sum of the angles in spherical triangles is greater than two right angles.    From Lambert, the story continues with the work of Legendre, Lobachewsky, Beltrami and Riemann. 

Dear Jose, your position is absolutely correct, the space of general relativity is not a flat space. But also the classical mechanics has an important place in physics (see my article on the truth) and then also the flat space of Euclidean geometry ghas an important place in mathematics. My question was limited to the flat geometry of Euclidean space.So your interesting observations are considered, for me, off topic.

Dear James, I thank you for the interesting comments. I merely observe that mainly concern the important concept of axiomatic theory. I take this opportunity to explain that

I have always been highly appreciative of James’s profound understanding of mathematics and its history and philosophy. He is quite right in pointing out that “It was Euclid of Alexandria who introduced a framework for doing mathematics, starting with definitions and axioms followed by theorems. It is observed (p. 5) that Euclid introduced the paradigm for reasoning and proving in mathematics that has endured for over 20 centuries into the present” without acknowledging contribution of ancient Egyptians who laid down foundation of geometry in the first place and Euclid and others built on that foundation. Since, question is related to Euclidean geometry, so he is right doing so. Of course, since Euclid first published his book Elements in 300 B.C. it has remained remarkably correct and accurate to real world situations faced on Earth. It is good for architecture and to survey land because it correctly describes physical space.

Gauss was one of the first to understand that the truth or otherwise of Euclidean geometry was a matter to be determined by experiment and he even went so far as to measure the angles of the triangle formed by three mountain peaks to see whether they added to 180. (Because of experimental error, the result was inconclusive). After Gauss, it was still reasonable to think that, although Euclidean geometry was not necessarily true (in the logical sense) it was still empirically true: after all, draw a triangle, cut it up and put the angles together and they will form a straight line. Our present-day understanding of models of axioms and so on can all be traced back to it.

The one problem that some find with it is that it is not accurate enough to represent the three dimensional universe that we live in. It has been argued that when it is moved into the third dimension, the postulates do not hold up. Non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one. Hyperbolical and spherical or elliptical geometry hold true in a two dimensional world, as well as in the third dimensional one.

The philosophical importance of non-Euclidean geometry is that it has greatly clarified the relationship between mathematics, science and observation. The scientific importance is that it has made way for the development of Riemannian geometry, which in turn paved the way for Einstein's General Theory of Relativity. After Einstein belief in exactness of Euclidean geometry might have been abandoned as it has been known that Euclidean geometry is only an approximation to the geometry of actual, physical space. This approximation is pretty good for everyday purposes, but would give bad answers in certain circumstances like near a black hole, if its existence still holds true as after rubbishing of theory of formation of black holes by Hawking himself, scientists are keeping it alive and research papers related to are are being published now and then.

As James has demonstrated within a two-dimensional plane, for any given line ℓ and a point A, which is not on ℓ, there is exactly one line through A that does not intersect ℓ. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting ℓ, while in elliptic geometry, any line through A intersects ℓ.

But, as he himself once pointed out to me that a bundle of parallel rays from infinity, will bend and will make loops and will not strike an object in a parallel fashion, but it ts possible in hyperbolic and elliptic geometries not Euclidean.

v

Dear @Mohammad Firoz Khan, many thanks for your incisive as well as gracious remarks.   I am indebted to you for the many good discussions that you inspired and which we have had on related topics in the past.

Also, I do agree with you that I should have taken into account the contributions made by Egyptian geometers. 

Apparently, Greek mathematicians (e.g., Pythagoras of Samos (572 B.C.E.-490 B.C.E.) and philosophers (e.g., Thales of Milet (624 B.C.E.-546 B.C.E.) were heavily influenced by Egyptian surveying skills and Egyptian geometry.    For more about this, see

T. Kotnik, Geometry and the Body, ETH Zurich:

http://www.schwartz.arch.ethz.ch/Publikationen/Dokumente/futureTraditions.pdf

Presently neither is there a single mathematics or geometry. There is no question of being Euclid good or bad for mathematics. In case problems in which his geometry can accurately solve them use it otherwise there are other geometries among  them whichever appears most suitable to solve a problem, can be used. All problems related to architecture/construction and land surveying in an area of 10 x 10 km area can be solved using Euclidean geometry, but as the length or width of an area increases 10 km, curvature of earth is to be taken into account and here spherical or ellipsoidal geometry/trigonometry is to be applied.