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Understanding Inequalities in Triangles: Relationships Between Angles and Sides

This lesson explores the fundamental principles of inequalities in triangles, focusing on the relationships between the sides and angles. You'll learn to identify and utilize the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. We will also cover examples of finding possible side lengths and understanding how to rank angles based on their measures. This foundational knowledge will aid you in solving triangle-related problems effectively.

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Understanding Inequalities in Triangles: Relationships Between Angles and Sides

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  1. Vocabulary Inequalities Within a Triangle What You'll Learn You will learn to identify the relationships between the _____ and _____ of a triangle. sides angles Nothing New!

  2. P 11 8 M 13 L Inequalities Within a Triangle in the same order PM < ML LP < mM < mL < mP

  3. K W 45° 75° 60° J Inequalities Within a Triangle in the same order mW < mJ < mK KW < WJ JK <

  4. W X Y Inequalities Within a Triangle greatest measure 5 3 4 WY > XW WY > XY

  5. Inequalities Within a Triangle The longest side is The largest angle is So, the largest angle is So, the longest side is

  6. B A C Triangle Inequality – examples… For the triangle, list the angles in order from least to greatest measure. 4 cm 6 cm 5 cm

  7. Vocabulary Triangle Inequality Theorem What You'll Learn You will learn to identify and use the Triangle Inequality Theorem. Nothing New!

  8. b a c Triangle Inequality Theorem greater a + b > c a + c > b b + c > a

  9. However, 10 + 5 > 16 Triangle Inequality Theorem Can 16, 10, and 5 be the measures of the sides of a triangle? 16 + 10 > 5 No! 16 + 5 > 10

  10. Can 11, 21, and 31 make a Triangle

  11. If two sides of a triangle are 7 & 13, between what two numbers must the third side be? -if 13 is the largest side then the smallest side had to be > 6 (13 < 7 + ? ) -If 7 & 13 are the 2 smaller sides, the 3rd side has to be < 20 (? < 13 + 7 ) Answer: the 3rd side is between 6 and 20. Example: Triangle Inequality

  12. Example Four:Finding Possible Side Lengths • A triangle has sides of lengths 8 and 10. Describe the lengths of the third side. • Let x represent the third side.

  13. Checkpoint

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