Which biconditional is not a good definition
a. two angles are supplementary if and only if the sum of their angles measures 180.
b. two angles are vertical angles only if they are not adjacent to each other and are formed by two intersecting lines.
c. two angles form linear pair if and only if the angles are adjacent. **
d. the sum of two angles is 90 if and only id those angles are complementary

The answer is: [C]:  Two angles form [a] linear pair if and only if the angles are adjacent.

{Your answer choice is correct! Good job!}.

Explanation:

Consider the answer choice given in: "Answer choice: [C]" :

"Two angles form linear pair if and only if the angles are adjacent."

       Then, consider the following:
 
        1)  "Two angles that are supplementary form a linear pair."
Is this true?   Yes, but not always. (In other words, "sometimes").  A "linear pair" of angles IS supplementary (by definition); however, NOT ALL supplementary pairs of angles are linear angles.  Supplementary pairs of angles must ALSO be ADJACENT in order to be "linear angles" as well. 
        2)  "Two angles that are vertical are adjacent."  Is this true?  No; this is NEVER true

As such, this biconditional is NOT a good definition; and by examining all the other answer choices given, we see that this answer choice "C" is correct.

Consider the other answer choices given:

Answer choice: [A]: "Two angles are supplementary if and only if the sum of their angles measures 180." ;
   is INCORRECT because it is a good definition:
        1)  If 2 angles are supplementary, then the sum of their measures is 180 degrees.
        2) If the sum of the measures of two angles is 180 degrees, then the angles are supplementary.

Answer choice: [B]: "Two angles are vertical angles only if they are not adjacent to each other and are formed by two intersecting lines."; 
    is INCORRECT because it is a good definition:
        1) If two angles are vertical, then they are nonadjacent angles formed by two intersecting lines.
        2) If two angles are nonadjacent angles formed by two intersecting lines, then the angles are vertical angles.

Answer choice: [D]: "The sum of two angles is 90 if and only if those angles are complementary" ; 
     is INCORRECT because it is a good definition: 
        1) If two angles are complementary, then the sum of their measures is 90 degrees.
        2) f the sum of [the measures of] two angles is 90 degrees, then the angles are complementary.

Answer

1 number

explanation

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

 

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.