How does King use The Constitution and reasoning to support his argument for civil rights?

Martin Luther King, Jr. composed this letter to highlight his reasons for defying a law that prevented demonstrations, and more importantly, to persuade "men of genuine good will" to actively fight injustice and support the civil rights movement. King addresses the letter to clergymen specifically, and appeals to readers through common sense, biblical principles, and references to history, including documents such as the U.S. Constitution.
King uses reasoning to highlight the hypocrisy of people who vehemently oppose demonstrations, yet care nothing about the inequality that inspired the demonstrations. He details instances of brutality and segregation in Birmingham, as well as a failure on the part of those in power to right such wrongs, pointing out that a recent mayoral election had not expedited change. In the following famous quote, King decries the logic shared by many that said he should wait for a more opportune moment to protest:

We know through painful experience that freedom is never voluntarily given by the oppressor; it must be demanded by the oppressed. Frankly, I have yet to engage in a direct action campaign that was "well timed" in the view of those who have not suffered unduly from the disease of segregation. For years now I have heard the word "Wait!" It rings in the ear of every Negro with piercing familiarity. This "Wait" has almost always meant "Never." We must come to see, with one of our distinguished jurists, that "justice too long delayed is justice denied."

This appeal is designed to convict those guilty of apathy toward the immediate plight of black Americans. King notes that black Americans have waited more than 340 years for constitutional, God-given rights, and asserts that man has a moral obligation to break unjust laws. Combining knowledge of government with current events and theology, King points out that "a just law is a man made code that squares with the moral law or the law of God. An unjust law is a code that is out of harmony with the moral law." This point is designed to directly appeal to clergymen readers, who likely value a strict moral code in addition to religious precepts. An additional plea cites Christian ethics and respect for the U.S. judicial system, stating that

"...Segregation is not only politically, economically and sociologically unsound, it is morally wrong and sinful. Paul Tillich has said that sin is separation. Is not segregation an existential expression of man's tragic separation, his awful estrangement, his terrible sinfulness? Thus it is that I can urge men to obey the 1954 decision of the Supreme Court, for it is morally right; and I can urge them to disobey segregation ordinances, for they are morally wrong.

King further caters to his audience by referencing civil disobedience demonstrated in the Bible by Shadrach, Meshach and Abednego; this is contrasted with the legally sanctioned atrocities commited by law-abiding citizens in Nazi Germany. The Apostle Paul, John Bunyan, Martin Luther, Abraham Lincoln, and many more religious and historical figures are then cited, as is the Declaration of Independence. King asserts that men will all behave in radical ways, then asks his audience to consider which path they will choose in the following quote:

Will we be extremists for hate or for love? Will we be extremists for the preservation of injustice or for the extension of justice? In that dramatic scene on Calvary's hill three men were crucified. We must never forget that all three were crucified for the same crime--the crime of extremism. Two were extremists for immorality, and thus fell below their environment. The other, Jesus Christ, was an extremist for love, truth and goodness, and thereby rose above his environment. Perhaps the South, the nation and the world are in dire need of creative extremists.

This letter is inspiring, well-loved by many and is often quoted, as it is beautifully penned and filled with wisdom and reproach. Regrettably, one could easily argue that the text is equally as relevant in the United States today as it was at the time of King's writing.