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?

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.