To clarify, the given conditional statement is:

"If two angles are adjacent, then they share a vertex."

The converse of a conditional statement reverses the "if" and "then" parts. The converse of this statement is:

"If two angles share a vertex, then they are adjacent."

To disprove the converse, we need a counterexample where two angles share a vertex but are not adjacent.

Counterexample:

Consider two angles that share a vertex but are on opposite sides of a common line, such as the angles formed by the hands of a clock at 12:00 and 6:00.

• Let’s say there is a point $O$, which is the common vertex of both angles.
• One angle is formed by rays $OA$ and $OB$, where $A$ and $B$ are points on opposite sides of $O$.
• Another angle is formed by rays $OA$ and $OC$, where $C$ is on the same side of the line as $B$.

These two angles share the vertex $O$, but they are not adjacent because they do not share a common side.

Thus, the counterexample demonstrates that the converse is false. Two angles can share a vertex but not be adjacent.

Would you like further details or clarifications?

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Here are 5 related questions:

1. What does it mean for two angles to be adjacent?
2. Can two adjacent angles be supplementary?
3. What are vertical angles, and how do they relate to this concept?
4. What is the difference between a conditional and its inverse?
5. How do you prove a converse statement true in general?

Tip: When working with conditionals and converses, always remember to test specific cases to find counterexamples or patterns.