Triangle inequalities and the relationship between sides and angles in...
Algebra 194 views·Updated Jul 7, 2026·2 pagesUnderstanding the Triangle Inequality Theorem
Eithar Omer@eitharomer
1
of 2
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 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.
2
of 2
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.
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Algebra 194 views·Updated Jul 7, 2026·2 pagesUnderstanding the Triangle Inequality Theorem
Eithar Omer@eitharomer
Triangle inequalities and the relationship between sides and angles in triangles are fundamental concepts in geometry. Understanding these relationships helps you determine whether given lengths can form a triangle and predict the order of sides and angles.
1
of 2
Sign up to see the content. It's free!
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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 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.
2
of 2Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
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.
We thought you’d never ask...
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
You can download the app in the Google Play Store and in the Apple App Store.
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