https://file.aiquickdraw.com/mathbot/file/1728675331625kz8051qg.jpg

Let me walk you through the answers for each of the problems shown in the image:

1. **Identify the hypothesis and conclusion below:**
   - The statement is: "The show is cancelled if it rains."
   - Hypothesis: **It rains.**
   - Conclusion: **The show is cancelled.**
   
2. **A converse statement is written in what form?**
   - The answer is **(b) If q, then p.**
   - A converse of "If p, then q" switches the hypothesis and conclusion, so it becomes "If q, then p."

3. **A conditional statement is written in what form?**
   - Your answer says "not q and not p," but this would describe the *inverse* of a conditional, not the conditional itself.
   - A conditional is written as: **If p, then q.**

4. **Write the converse of the following conditional statement:**
   - Conditional: "If an angle is obtuse, then it has a measure of 100 degrees."
   - Converse: **If an angle has a measure of 100 degrees, then it is obtuse.**
   - The truth value: This is **false** because an obtuse angle is greater than 90 degrees but less than 180 degrees, and 100 degrees is just one example of an obtuse angle.

5. **Write the inverse of "If two angles are vertical angles, then they are congruent":**
   - Inverse: **If two angles are not vertical angles, then they are not congruent.**

6. **If AB = BC and BC = CD, then AB = CD. What type of property?**
   - This is an example of the **Transitive Property**.

7. **What does "iff" mean?**
   - "Iff" means **if and only if**, a term used in logical statements indicating that both conditions must be true or both false.

8. **If AB = CD, then CD = AB. What property is shown?**
   - This is an example of the **Symmetric Property**.

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Let me know if you'd like any further clarifications or details on any specific topic!

Here are some additional related questions for you to consider:

1. What is the contrapositive of a conditional statement?
2. How do you determine the truth value of a converse?
3. How would you write the contrapositive of the statement "If it rains, then the show is cancelled"?
4. Can a converse statement always be true if the original conditional is true?
5. How can the Transitive Property be applied in real-world situations?

**Tip:** Always remember that the converse, inverse, and contrapositive of a conditional statement are different, and their truth values may not always match the original statement.

Identify the hypothesis and conclusion, conditional statements, converse, inverse, and mathematical properties in logical statements.

Learn how to identify the hypothesis, conclusion, and forms of conditional statements. Explore the converse, inverse, and logical properties such as Transitive and Symmetric Property in geometry, suitable for high school students.

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