To solve this problem, we need to order the side lengths of triangle $\triangle HIJ$ from least to greatest. The triangle has angles:

• $\angle H = 85^\circ$
• $\angle J = 44^\circ$ (since the triangle's angles must sum to 180 degrees, $180 - 85 - 51 = 44$)
• $\angle I = 51^\circ$

In a triangle, the side opposite the largest angle is the longest, and the side opposite the smallest angle is the shortest. The side lengths to compare are:

1. $HI$ opposite $\angle J = 44^\circ$
2. $IJ$ opposite $\angle H = 85^\circ$
3. $JH$ opposite $\angle I = 51^\circ$

Ordering the sides by the angles they are opposite to:

• $IJ$ is the longest (opposite the largest angle, $85^\circ$).
• $JH$ is next (opposite the middle angle, $51^\circ$).
• $HI$ is the shortest (opposite the smallest angle, $44^\circ$).

Thus, the order of the side lengths from least to greatest is:

$HI < JH < IJ$

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

Related Questions:

1. How can we use trigonometric ratios to find exact side lengths of a triangle given angles and one side length?
2. What is the significance of the Triangle Inequality Theorem in determining side lengths?
3. How do different types of triangles (like isosceles or equilateral) affect the relationships between angles and sides?
4. How does knowing two angles of a triangle help us determine the third?
5. What other properties of triangles are important in geometry?

Tip:

When comparing sides of a triangle, always look at the angles opposite those sides. The largest angle will have the longest side opposite it, and the smallest angle will have the shortest side.