Key ways the Koran and the Book of Mormon are alike in the way they relate to the Bible and to each other.

The Koran and the Book of Mormon share several common features.  This fact was not lost on nineteenth-century Americans who often referred to Joseph Smith Jr. as the American Mohammad.
  

Both books were the product of a single individual (one ostensibly translating from golden plates and the other taking down dictation form the mouth  of the Angel Gabriel).

Both ultimately stand of fall on the validity of the claims of a single prophetic author.   This is in marked contrast to the Bible, which was written over many centuries by a multiplicity of prophetic authors.

Both bear the same relation to the Bible, namely they both represent themselves as the final step in the flow of biblical revelation, the third “Testament,” as it were, supplementing the Old and New Testaments and bringing divine revelation to a final, divinely authoritative culmination.  

Both claim to serve as confirmation of the Bible. We see this, for example, in the Book of Mormon’s description of itself as “Another Testament of Jesus Christ,” and the Koran’s claim that “a Book confirming their own [i.e., the Bible] has come to them from God” (Surah 2:89).

Both explain (and/or their defenders explain) that contradictions between their teachings and the teaching of the Bible result from no flaw in themselves but rather from the supposedly radical corruption of the Biblical text (This in contrast to the Christian tradition which, since the Marcionite controversy of the 2nd century, has insisted that the Old Testament enjoys equal status with the New). 

[It has to be said as an aside, however, that the Koran adopts a much more consciously strident adversarial tone in relation to the Bible's teaching than does the Book of Mormon. Indeed, readers of the Bible who then turn to the Koran may very easily conclude that the whole purpose for the Koran's being written, its very raison d'être, was to confute and contradict the view of salvation history represented in the Bible, and most particularly the New Testament account of what Jesus and his earliest followers said about who he was and what he had come in to this world to accomplish.]

Both credit resistance toward accepting their explanations of differences and contradictions between them and the Bible to hard-hearted spiritual intransigence on the part of those doing represented in an unwillingness to keep in step with God’s ongoing revelation.  So we read in the Book of Mormon, 2 Nephi 29:3:

“…many of the Gentiles shall say: A Bible! A Bible! We have got a Bible, and there cannot be any more Bible...Thou fool, that shall say: A Bible, we have got a Bible, and we need no more Bible!”     

And in the Koran, Surah 2:91:

“And if they are told: ‘Believe in what God has revealed,’ they reply: ‘We believe in what was revealed to us.’ And they deny what has since been revealed, although it is the Truth, corroborating their own Scriptures.”

Both books follow the Bible in embracing the concept of religious exclusivism, an idea that rules out the possibility of affirming that all religions are the same, that all are true, that all can be counted on to equally lead us to God as various paths might lead us to the top the same mountain, and so on.  Each book makes claims for it’s own validity that rules out the possible validity of the claims of the other.  Hence, the claims of the Book of Mormon and the Koran may both be false, but they cannot both be true.  Its not my purpose to enter into that debate here.  But the answer as to which is true and which false will naturally seem obvious to believers in the one or the other.