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This worksheet covers inequalities in triangles, specifically focusing on the relationships between angles and sides within triangles. Let’s break down the key exercises and how to approach them:

1. **Explanation of Inequalities in One Triangle**:
   - The first part mentions the relationship between the measure of an exterior angle of a triangle and the two remote interior angles.
   - The exterior angle is always greater than either of its remote interior angles, meaning that for a triangle with angles \(m \angle 1\), \(m \angle 2\), and \(m \angle 3\), the exterior angle (here it seems labeled as \(m \angle 1\)) is greater than the other two, i.e., \(m \angle 1 > m \angle 2\) and \(m \angle 1 > m \angle 3\).

2. **Exercises 2-5**: You are asked to list the angles of each triangle in order from smallest to largest based on the given side lengths. Here’s the approach:
   - **Exercise 2 (Triangle ABC)**: Use the sides provided (3.1, 4.4, 2.7). The smallest angle is opposite the shortest side, and the largest angle is opposite the longest side.
     - Since side AC is the shortest (2.7), the angle opposite it (∠B) will be the smallest.
     - Since side AB is the longest (4.4), the angle opposite it (∠C) will be the largest.
   - **Exercises 3-5**: Follow the same approach, where the smallest angle corresponds to the smallest side, and the largest angle corresponds to the longest side.

3. **Exercise 6-7**: In these triangles, you are given angle measures. To order the sides from shortest to longest, the smallest side is opposite the smallest angle, and the longest side is opposite the largest angle.
   - For example, in Exercise 6, ∠P = 90°, ∠Q = 50°, and ∠R = 40°, so:
     - The longest side is opposite the 90° angle (side PR),
     - The next longest side is opposite the 50° angle (side PQ),
     - The shortest side is opposite the 40° angle (side QR).

4. **Exercise 8**: You are given three angles (∠M = 56°, ∠N = 108°, and ∠O = 16°). The task is to list the sides in order from shortest to longest based on the angle measures. Remember, the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle.

5. **Exercise 9 (Algebra)**: You are asked to list the sides in order based on the angles provided in terms of \(x\):
   - You have ∠X = 50°, ∠Y = 5x + 10, and ∠Z = 5x.
   - You would first solve for \(x\) by using the triangle sum theorem (i.e., the sum of the angles in any triangle is 180°), then compare the angles to order the sides.

Would you like to go into further detail on any of these exercises? Let me know if you'd like more guidance, and I'll be happy to provide a step-by-step solution.

### Related Questions:
1. How do you solve for \(x\) in Exercise 9 using the triangle sum theorem?
2. Why is the exterior angle of a triangle always greater than the remote interior angles?
3. What is the significance of the relationship between angles and side lengths in any triangle?
4. How can you calculate the third angle of a triangle if two angles are given?
5. In what cases would a triangle be classified as scalene, based on its sides and angles?

**Tip**: Always remember that in any triangle, the side opposite the largest angle is the longest, and the side opposite the smallest angle is the shortest.

The user submitted an image of a worksheet focusing on inequalities in triangles, including relationships between angles and sides, ordering angles and sides, and some algebra-based problems involving triangles.

Learn how to solve problems involving inequalities in triangles with this practice worksheet. Explore key concepts like the exterior angle theorem, triangle sum theorem, and ordering sides and angles based on inequalities. Ideal for students in Grades 8-10.

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