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Let's analyze and solve the given geometric problem:

### Part (a) - Ordering the Angle Measures
We are asked to order the angles \( m\angle G \), \( m\angle H \), and \( m\angle GIH \) from least to greatest.

From the diagram:
- \( \angle GIH = 62^\circ \) (given).
- The other two angles, \( \angle G \) and \( \angle H \), belong to triangle \( GHI \), but they are not directly given. However, we can use the **Triangle Sum Theorem**, which states that the sum of the interior angles of a triangle is always \( 180^\circ \). So:
  \[
  m\angle G + m\angle H + m\angle GIH = 180^\circ
  \]
  Plugging in the known value of \( m\angle GIH = 62^\circ \), we get:
  \[
  m\angle G + m\angle H = 180^\circ - 62^\circ = 118^\circ
  \]
  To determine the individual values of \( \angle G \) and \( \angle H \), we can apply the **Law of Sines** (or recognize that the larger side corresponds to the larger opposite angle). Since side \( GH = 11.12 \), \( G \) will have a larger angle than \( H \), and side \( GI = 9 \), opposite \( \angle H \), will correspond to a smaller angle. Thus, we can approximate the angles, with \( \angle G \) being larger than \( \angle H \).

Thus, the order of angles is:
\[
m\angle H < m\angle G < m\angle GIH
\]

### Part (b) - Ordering the Side Lengths
We need to order the side lengths \( IJ \), \( JK \), and \( IK \).

From the diagram:
- \( IJ \), \( JK \), and \( IK \) are the sides of triangle \( IJK \), where \( \angle J = 65^\circ \) and \( \angle I = 62^\circ \).
- Using the **Triangle Angle-Side Relationship**, the side opposite the largest angle is the longest. Since \( \angle K \) will be the smallest angle (since \( \angle J + \angle I = 127^\circ \), so \( \angle K = 180^\circ - 127^\circ = 53^\circ \)), we know:
  \[
  IJ < IK < JK
  \]
  because \( \angle J > \angle I > \angle K \).

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Would you like further details or additional clarifications?

Here are 5 questions to deepen understanding:
1. How does the Triangle Sum Theorem apply in different types of triangles?
2. What is the significance of the Law of Sines in solving triangle problems?
3. How can you confirm the order of angles and sides without measuring them directly?
4. What other methods could be used to solve for unknown angles in a triangle?
5. How does the Triangle Angle-Side Relationship help in comparing sides of a triangle?

**Tip:** Always check if you can use angle or side relationships before applying trigonometric laws. This can simplify problems significantly.

Order the angle measures m∠G, m∠H, and m∠GIH from least to greatest. Order the side lengths IJ, JK, and IK from least to greatest.

Solve the problem of ordering the angle measures m∠G, m∠H, and m∠GIH, and side lengths IJ, JK, and IK from least to greatest. Explore key concepts like the Triangle Sum Theorem and Law of Sines in this geometry problem, ideal for students in Grades 9-11.

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