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### 7. Two planes intersect at a line.
This statement is true based on geometry. The **if-then form** is:  
**If** two planes intersect, **then** they intersect at a line.

- **Converse**: If two planes intersect at a line, then they are two planes (True).
- **Inverse**: If two planes do not intersect, then they do not intersect at a line (True).
- **Contrapositive**: If two planes do not intersect at a line, then they do not intersect (True).
- **Biconditional**: Two planes intersect if and only if they intersect at a line (True).

### 8. A relation that pairs each input with exactly one output is a function.
This is true by definition of a function. The **if-then form**:  
**If** a relation pairs each input with exactly one output, **then** it is a function.

- **Converse**: If a relation is a function, then it pairs each input with exactly one output (True).
- **Inverse**: If a relation does not pair each input with exactly one output, then it is not a function (True).
- **Contrapositive**: If a relation is not a function, then it does not pair each input with exactly one output (True).
- **Biconditional**: A relation is a function if and only if it pairs each input with exactly one output (True).

### 9. Use inductive reasoning to make a conjecture about the sum of three odd integers. Then use deductive reasoning to show that the conjecture is true.
- **Inductive Reasoning**: 
  - Example: \(3 + 5 + 7 = 15\), which is odd.
  - Example: \(1 + 3 + 5 = 9\), which is odd.
  - Conjecture: The sum of three odd integers is always odd.

- **Deductive Reasoning**: 
  Let three odd integers be represented as \(2n_1 + 1\), \(2n_2 + 1\), and \(2n_3 + 1\), where \(n_1, n_2, n_3\) are integers.
  \[
  (2n_1 + 1) + (2n_2 + 1) + (2n_3 + 1) = 2(n_1 + n_2 + n_3) + 3
  \]
  Since \(2(n_1 + n_2 + n_3)\) is even, adding 3 results in an odd number. Thus, the conjecture is true.

### 10. Apply the Law of Detachment
The Law of Detachment states that if a conditional statement is true and its hypothesis is true, then the conclusion is true. 

- Given:  
  "If a figure is a rectangle, then it has four sides" (True).  
  "Quadrilateral ABCD has four sides" (True).

However, you cannot conclude that quadrilateral ABCD is a rectangle solely based on the fact that it has four sides, because the conclusion must come from the hypothesis. Hence, the Law of Detachment **does not apply** here.

### 11. Solve for \( h \) and find the height of a standard yield sign.
The formula for the area of a triangle is given by:
\[
A = \frac{1}{2}bh
\]
Solve for \( h \):
\[
A = \frac{1}{2}bh \implies h = \frac{2A}{b}
\]
Given the area \( A = 558 \) square inches and the base \( b = 36 \) inches:
\[
h = \frac{2(558)}{36} = \frac{1116}{36} = 31 \text{ inches}
\]
Thus, the height \( h \) is 31 inches.

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Do you have any questions or would you like more details on any of these problems?

Here are five related questions:
1. How does the Law of Syllogism relate to the Law of Detachment?
2. Can the sum of an even number of odd integers be even?
3. What are the differences between inductive and deductive reasoning?
4. How would the Law of Detachment apply if more specific details were given about quadrilateral ABCD?
5. What are the possible dimensions of a triangle given its area and base?

**Tip**: Always check the hypothesis carefully when applying the Law of Detachment!

Write the if-then form, the converse, the inverse, the contrapositive, and the biconditional of the conditional statement. Then decide whether each statement is true or false.

Explore the breakdown of conditional statements and their forms, including the if-then form, converse, inverse, and contrapositive. Learn how to apply the Law of Detachment and use inductive and deductive reasoning to solve geometry problems. Suitable for high school students in Grades 9-12.

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