Triangle Inequality Theorem

Ever wonder if three random lengths can form a triangle? Here's the rule: a triangle exists when the sum of the two shorter sides is greater than the longest side. This is the Triangle Inequality Theorem.

To check if lengths can form a triangle, add the two smaller numbers and compare to the third. For example, with sides 8, 17, and 24, we check if 8+17>24. Since 25>24, these lengths can form a triangle. But for sides 3, 3, and 7, we see that 3+3=6 which is less than 7, so no triangle is possible.

When you know two sides of a triangle, you can determine the possible range for the third side. The third side must be greater than the difference of the two sides and less than their sum. For instance, if two sides are 14 and 22, then the third side $x$ must satisfy: |14-22| < x < 14+22, which means 8 < x < 36.

Remember This! When checking if lengths can form a triangle, you must verify the inequality for all three combinations of sides, though typically checking the sum of the two smaller sides against the longest is sufficient.

Triangle Side-Angle Relationships

Did you know the sizes of angles in a triangle directly relate to the lengths of the sides? This relationship works both ways and makes triangles predictable.

The smallest angle in a triangle is always opposite the shortest side. Similarly, the largest angle is always opposite the longest side. For example, in a triangle with sides 7 ft, 10 ft, and 13 ft, the smallest angle is opposite the 7 ft side, and the largest angle is opposite the 13 ft side.

When ordering angles in a triangle, look at the sides opposite to them. If the sides are 17m, 20m, and 24m, the angles would be ordered from smallest to largest based on the sides they face.

Quick Tip: Draw a quick sketch when solving these problems. Labeling the sides and angles makes it much easier to see the relationships and avoid mix-ups.

Working backward works too! If you know the angle measures, you can order the sides. The smallest side is always opposite the smallest angle, and the largest side is opposite the largest angle. This relationship helps you analyze triangles even when you don't have all the measurements.