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- Describe the relative size of the angle that is opposite the longest side.
- Describe the relative size of the angle that is opposite the shortest side.
- If you know the measure of the angles, but not the sides, how could you order the sides by length?
- If you know the lengths of the sides, but not the measure of the angles, how could you order the angles by size?

For a triangle to exist (be considered **valid**), the length of each side must be within a range that is determined by the lengths of the other two sides.

We can make a connection between the relative size of each angle and its opposite side length in a triangle. The smallest angle will be opposite the shortest side, while the largest angle will be opposite the longest side of the triangle.

Since the hypotenuse of a right triangle is always the longest side and is opposite the right angle, the legs must be shorter and opposite acute angles using the side-angle relationship.

Suppose that we have three sides of lengths 4, 7 and 12. Determine if these three sides can form a valid triangle.

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For the triangle in the figure, state its angles in order of ascending measure, from smallest to largest.

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For the triangle in the figure, order the sides from shortest to longest.

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A valid triangle has side lengths of 4, 10 and x.

Find the range of values for x.

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Minerva is hiking in Redwood National Park. She is camping in Berry Glenn and plans to hike up to Orick Hill. After the hike, she takes the trail down to Orick. Will she need to hike further than 2.5 \text{ miles}?

Worked Solution

Idea summary

Triangles are valid if they satisfy the **triangle inequality theorem**, which states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

The smallest angle in a triangle will be opposite the shortest side, while the largest angle will be opposite the longest side of the triangle.