https://file.aiquickdraw.com/mathbot/file/1727460177047gk2h7va9.jpg

The problem asks to determine the truth value of the conjunction "p and ~q" based on the given statements and the triangle shown in the image.

- **p**: The triangle has at least two congruent sides.
- **q**: The triangle has an obtuse angle.

The compound statement "p and ~q" means that the triangle has at least two congruent sides and does **not** have an obtuse angle.

### Analysis:

- The triangle shown in the image is a right triangle, not an obtuse triangle (since it has a right angle, which is 90 degrees). Thus, the statement **q** ("The triangle has an obtuse angle") is false. Therefore, **~q** (not q) is **true**.
- The triangle has two congruent sides (the two legs of the right triangle are marked as congruent). Therefore, **p** is **true**.

Since both **p** is true and **~q** is true, the conjunction **p and ~q** is **true**.

### Conclusion:
The correct answer is **B**: "The triangle has at least two congruent sides and does not have an obtuse angle" with a truth value of **True**.

Would you like more details on this, or do you have any questions?

Here are 5 related questions:
1. What is a conjunction in logic, and how is its truth value determined?
2. How does a right triangle differ from an obtuse triangle?
3. What are congruent sides, and how are they identified in a triangle?
4. Can a triangle have more than one obtuse angle? Why or why not?
5. How does the presence of a right angle affect the classification of a triangle?

**Tip**: In logic, the truth of a conjunction depends on both statements being true. If either is false, the entire conjunction is false.

Use the following statements to write a compound statement for the conjunction p and ~q. Then find its truth value.

p: The triangle has at least two congruent sides.
q: The triangle has an obtuse angle.

Explore how to determine the truth value of a logical conjunction involving geometric properties of a triangle. This problem involves understanding triangle types, congruent sides, and logical statements. Suitable for students in Grades 9-10.

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