Let’s analyze the question in your uploaded image. The task is to determine which of the given statements has a false converse. To do this, we will evaluate the converse of each statement.

1. Definitions:

• Original statement: A → B (if A, then B).
• Converse: B → A (if B, then A).

To check if the converse is false, we need a counterexample where B is true but A is false.

Analysis of Each Option:

A. If m∠A = 90, then ∠A is a right angle.

• Converse: If ∠A is a right angle, then m∠A = 90.
• The converse is true because by definition, a right angle always measures 90°.
• Not the answer.

B. If a point bisects a segment into two congruent parts, then it is a midpoint.

• Converse: If a point is a midpoint, then it bisects a segment into two congruent parts.
• The converse is true because by definition, a midpoint always bisects a segment into two congruent parts.
• Not the answer.

C. 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 two angles whose sum is 90° are, by definition, complementary.
• Not the answer.

D. If two angles are vertical, then they are congruent.

• Converse: If two angles are congruent, then they are vertical.
• The converse is false, because two angles can be congruent without being vertical (e.g., two corresponding angles in parallel lines cut by a transversal).
• Answer: D.

Final Answer:

D. If two angles are vertical, then they are congruent.

Would you like further clarification or detailed examples for any of the steps?

5 Related Questions:

1. What is the difference between a statement and its converse in logic?
2. How can you identify whether a converse of a statement is true or false?
3. What are examples of statements whose converses are always true?
4. Can a converse and an inverse of a statement both be false?
5. How do logical relationships like contrapositive and converse differ?

Tip:

When analyzing converses, try testing with counterexamples to check if the converse fails under certain conditions.