ABCD is a quadrilateral such that AB is the largest and CD is the smallest side of the quadrilateral. Then, which of the following option(s) is/are true?

∠A > ∠C

∠A < ∠C

<B> <D

<B< ∠D

Given the quadrilateral \(ABCD\) where \(AB\) is the largest side and \(CD\) is the smallest side, we need to determine which of the given options are true:

### Understanding the Angles:
- In any quadrilateral, the larger angle is opposite the longer side, and the smaller angle is opposite the shorter side.

Given that \(AB\) is the largest side:
- The angle opposite to \(AB\) is \(\angle C\).

Given that \(CD\) is the smallest side:
- The angle opposite to \(CD\) is \(\angle A\).

### Analyzing the Options:
1. **Option 1: \(\angle A > \angle C\)**  
   Since \(AB\) is the largest side and \(\angle C\) is opposite to it, \(\angle C\) should be larger than \(\angle A\) (which is opposite the smallest side \(CD\)). Hence, this option is **false**.

2. **Option 2: \(\angle A < \angle C\)**  
   Since \(\angle C\) is opposite the largest side \(AB\), it should be larger than \(\angle A\). Therefore, this option is **true**.

3. **Option 3: \(\angle B < \angle D\)**  
   We are not provided with information about the relative lengths of \(BC\) and \(DA\), so we cannot directly conclude the relationship between \(\angle B\) and \(\angle D\). This option is **indeterminate** based on the given information.

4. **Option 4: \(\angle B < \angle D\)**  
   This option repeats the previous one, so the same reasoning applies. This option is **indeterminate**.

### Conclusion:
The only true statement based on the given information is:

- **\(\angle A < \angle C\)**

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

### Follow-up Questions:
1. How does the sum of the angles in a quadrilateral relate to the side lengths?
2. Can we determine the relationship between \(\angle B\) and \(\angle D\) with additional information?
3. What if the quadrilateral were cyclic? How would that change the relationships?
4. How would the relationships change if \(AB\) and \(CD\) were adjacent instead of opposite?
5. How does this problem change if we consider it in the context of a parallelogram?

**Tip:** When analyzing a quadrilateral, always remember that larger sides generally have larger opposite angles.

Explore the relationships between angles in quadrilaterals where AB is the largest side and CD is the smallest side. Understand how to determine the relative sizes of angles A and C based on side lengths. Suitable for advanced high school students.

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