Is the converse of a false conditional always true as in the Truth Table? Accroding to the Truth Table, 
If $p$ is TRUE, and $q$ is FALSE, then $p\implies q$ is FALSE.
And the converse, $q \implies p$, is TRUE.
If the conditional statement is 
"If two angles are congruent, they are not equal."
this is a FALSE statement.
Its converse is 
"If two angles are not equal, they are congruent."
The converse is also FALSE.
Why does this example contradict the Truth Table??
 A: The source of the confusion is that you are not dealing merely with implications in propositional logic, where, as you said, the truth tables ensure that $(p\implies q)\lor(q\implies p)$ is always true.  Your statements about angles are universally quantified statements, even though the English language lets you hide the quantifiers.  Your first statement really means "For every two angles $x$ and $y$, if $x$ and $y$ are congruent then $x$ and $y$ are not equal."  Similarly for the second statement. 
So the logical form of these statements is $(\forall x)(\forall y)\,(P(x,y)\implies Q(x,y))$ and $(\forall x)(\forall y)\,(Q(x,y)\implies P(x,y))$.  Because of the quantifiers, it is entirely possible for both of these to be false.  All that's needed for that to happen is that some particular $x_0$ and $y_0$ satisfy $P$ but not $Q$, while a different pair $x_1$ and $y_1$ satisfy $Q$ but not $P$.
If quantifiers are not involved, for example if you have two particular angles and are making statements about just this pair, not angles in general, then a proposition and its converse will not both be false.
A: It doesn't. In your example let $q = $"two angles are not equal" be false so that means they are equal. Hence the statement "If two angles are not equal, they are congruent" is vacuously true.
