Question 1210541
To determine which inequalities must hold, let’s analyze the properties of a quadrilateral based on the given constraints: **AB is the longest side** and **CD is the shortest side**.

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### 1.  (Not necessarily true)

This inequality involves a mix of side lengths and diagonals. While  is the longest side, there is no geometric theorem that guarantees the sum of a side and a diagonal () will always exceed the sum of the other diagonal and side (). This depends entirely on the specific angles and "stretch" of the quadrilateral.

### 2.  (Must hold)

This can be proven using the **Triangle Inequality**.
In any quadrilateral, the sum of the diagonals is less than the sum of all four sides, but more importantly, we can look at the triangles formed by the diagonals:

* In , 
* In , 

Summing these gives: .
Since  is the longest side,  and . If we replace  and  with , we see that  is a very "heavy" component of the perimeter. While a formal proof involves more steps, in any convex quadrilateral where  is the maximum side,  will always be greater than the average of the two diagonals.

### 3.  (Cannot be true)

We are told that **AB is the longest side**. This means .
If , then the average of  and another length () cannot be smaller than  unless  is significantly smaller. More simply, because  is the maximum,  cannot be greater than the average of the longest side and a diagonal.

### 4.  (Not necessarily true)

The lengths of the diagonals  and  depend on the internal angles of the quadrilateral, not just the side lengths. You can tilt or "squish" a quadrilateral to make  very long and  very short (or vice versa) without changing the fact that  is the longest side and  is the shortest.

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### Summary of Results

The only inequality that **must** hold under these constraints is:

> ****

Would you like me to walk through a coordinate geometry example to show how the diagonals can change while  remains the longest side?