Sure, let's go through each question one by one:
1. **Which statement explains the Pythagorean Theorem?**
- *Correct Answer:* If a triangle is a right triangle, then the sum of the squares of the legs is equal to the square of the hypotenuse.
2. **Does the Pythagorean Theorem apply to this triangle? How do you know?**
- *Correct Answer:* Yes, because it is a right triangle. (As the triangle has a 90° angle at point F)
3. **Using the Converse of the Pythagorean Theorem, explain whether a triangle with sides 15, 10, and 12 is a right triangle.**
- Let's check \(15^2 = 225\), \(10^2 = 100\), and \(12^2 = 144\). Now, \(15^2\) does not equal \(10^2 + 12^2\) or \(225 \neq 144 + 100 = 244\).
- *Correct Answer:* No, because 244 is not equal to 225.
4. **If the equation \(18^2 + b^2 = 30^2\) is found to be true, what do we know about the triangle?**
- Solve for \(b\):
- \(18^2 + b^2 = 30^2\)
- \(324 + b^2 = 900\)
- \(b^2 = 900 - 324\)
- \(b^2 = 576\)
- \(b = 24\)
- *Correct Answer:* The triangle is a right triangle with a missing leg length of 24.
5. **What is a converse of a theorem that is an if-then statement?**
- *Correct Answer:* The if part and the then part switch places.
These selections ensure a clear understanding of the Pythagorean Theorem, its converse, and the conditions under which they apply.