Given the statements: - $A B \leq C$ - $B = D$ - $E \leq F \leq D$ Determine which of the following conclusions definitely follow: I. $E = C$ II. $C E$ III. $A B$

Question

Given the statements:

- $A > B \leq C$
- $B = D$
- $E \leq F \leq D$

Determine which of the following conclusions definitely follow:

I. $E = C$

II. $C > E$

III. $A < D$

IV. $F > B$

Accepted Answer

**Solution:**

1. From the given statements, we have:
   - $A > B \leq C$
   - $B = D$
   - $E \leq F \leq D$

2. Combining these, we get:
   - $A > B = D \leq C$
   - $E \leq F \leq D = B$

3. Analyze each conclusion:

- **Conclusion I: $E = C$**
     - Since $E \leq F \leq D = B \leq C$, $E$ can be equal to $C$ but not necessarily. So this conclusion may be true but not definite.

- **Conclusion II: $C > E$**
     - Given the chain $E \leq F \leq D = B \leq C$, $C$ can be greater than or equal to $E$. So this conclusion may also be true but not definite.

- **Conclusion III: $A < D$**
     - Since $A > B = D$, $A$ is greater than $D$, so this conclusion is false.

- **Conclusion IV: $F > B$**
     - Since $F \leq D = B$, $F$ cannot be greater than $B$, so this conclusion is false.

4. Therefore, only conclusions I or II may follow, but neither is definitely true.

**Answer:** Either I or II follows.

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