To solve this, we need to identify which of the given statements has a false converse. The converse of a conditional statement "If P, then Q" is "If Q, then P." Let's examine each statement and its converse.

Statement A:

Original: "If $m∠A = 90$, then ∠A is a right angle."
Converse: "If ∠A is a right angle, then $m∠A = 90$."
The converse of this statement is true because, by definition, a right angle always measures 90 degrees.

Statement B:

Original: "If a point bisects a segment into two congruent parts, then it is the midpoint."
Converse: "If a point is the midpoint, then it bisects the segment into two congruent parts."
The converse is true because the midpoint always divides a segment into two equal parts.

Statement C:

Original: "If two angles are a linear pair, then they are supplementary."
Converse: "If two angles are supplementary, then they are a linear pair."
This converse is false. Two angles can be supplementary (i.e., sum to 180°) without forming a linear pair. For example, two angles inside a triangle can be supplementary, but they are not a linear pair.

Statement D:

Original: "If two angles are complementary, then the sum of their measures is 90°."
Converse: "If the sum of two angles is 90°, then they are complementary."
The converse is true because by definition, complementary angles add up to 90°.

Conclusion:

The statement with a false converse is Statement C. The false converse is: "If two angles are supplementary, then they are a linear pair."

Would you like more details or further clarifications?

Related Questions:

1. What is the difference between a conditional statement and its converse?
2. Can a converse of a true statement ever be false, and if so, why?
3. How do supplementary angles differ from a linear pair of angles?
4. What are contrapositive and inverse statements in logic?
5. Can you give examples of real-life situations involving linear pairs or supplementary angles?

Tip:

Remember, a converse can be false even if the original statement is true. Always analyze the converse separately from the original statement.