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To determine the order of the side lengths in \( \triangle BCD \) from smallest to largest, we can analyze the triangle based on the angles provided.

### Step 1: Identify the Angles
The triangle \( \triangle BCD \) has the following angles:
- \( \angle B = u + 64^\circ \)
- \( \angle C = 55u \)
- \( \angle D = u + 59^\circ \)

Since the sum of the angles in a triangle must equal \( 180^\circ \), we can set up an equation to solve for \( u \):
\[
(u + 64) + (55u) + (u + 59) = 180
\]

### Step 2: Solve for \( u \)
1. Combine like terms:
   \[
   57u + 123 = 180
   \]
2. Subtract 123 from both sides:
   \[
   57u = 57
   \]
3. Divide by 57:
   \[
   u = 1
   \]

### Step 3: Substitute \( u = 1 \) Back into the Angles
Now that we know \( u = 1 \), substitute this value into each angle:
- \( \angle B = 1 + 64 = 65^\circ \)
- \( \angle C = 55 \cdot 1 = 55^\circ \)
- \( \angle D = 1 + 59 = 60^\circ \)

### Step 4: Determine Side Lengths Using Angle-Side Relationships
In any triangle, the side opposite the largest angle is the longest, and the side opposite the smallest angle is the shortest.

From our calculations:
- The largest angle is \( \angle B = 65^\circ \), so the side opposite \( B \) (which is \( \overline{CD} \)) is the longest.
- The smallest angle is \( \angle C = 55^\circ \), so the side opposite \( C \) (which is \( \overline{BD} \)) is the shortest.
- The remaining side \( \overline{BC} \), opposite \( \angle D = 60^\circ \), is of medium length.

### Conclusion
The order of the side lengths from smallest to largest is:
\[
\overline{BD} < \overline{BC} < \overline{CD}
\]

Would you like further details on any part of this solution?

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### Related Questions:
1. How can the Law of Sines help in determining side lengths in a triangle?
2. What is the relationship between the angles and side lengths in any triangle?
3. How do you verify if three given angles can form a triangle?
4. What is the significance of finding angle measures in non-right triangles?
5. How would the order of side lengths change if one of the angle measures increased?

### Tip:
In any triangle, the side opposite the largest angle is always the longest side, and the side opposite the smallest angle is the shortest.

List the side lengths of ΔBCD in order from smallest to largest.

Learn how to order the side lengths of triangle ΔBCD by comparing the angle measures. This problem demonstrates the angle-side relationship in triangles and the steps to solve for unknown values. Suitable for students in Grades 8-10.

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