Lesson Description
This lesson provides a structured problem set on triangle inequalities. Students will determine whether three segment lengths can form a triangle, compare angles and opposite sides, and apply inequality relationships in geometric proofs.
Learning Objectives
- Apply the Triangle Inequality Theorem.
- Determine if three lengths form a valid triangle.
- Compare side lengths using angle measures.
- Use inequalities in geometric reasoning.
Topics Covered
- Triangle Inequality Theorem: \(a+b>c\)
- Longest side opposite largest angle
- Shortest side opposite smallest angle
- Perimeter bounds using inequalities
Key Terms
- Triangle Inequality Theorem: The sum of any two sides must exceed the third.
- Opposite Side: The side across from a given angle.
- Exterior Inequality: Exterior angles exceed either remote interior angle.
Practice Problems
- Determine if a triangle can exist with sides:
- 7, 10, 5
- 3, 4, 9
- 12, 15, 20
- Triangle ABC has angles \(A=40^\circ\), \(B=65^\circ\), \(C=75^\circ\). Order sides \(a,b,c\) from shortest to longest.
- Two sides of a triangle are 8 and 11. Find the possible integer values for the third side.
- A triangle has sides 6, 8, and \(x\). Write the inequality describing all possible values of \(x\).
- If the longest side of a triangle is 18 and another side is 7, what is the minimum integer length of the third side?
Challenge Problems
- A triangle has perimeter 30. Two sides measure 9 and 12. Find the range of the third side.
- Prove that the straight-line distance between two points is always the shortest path using triangle inequalities.
- Three segments measure \(x+2\), \(x+5\), and 20. Find all integer values of \(x\) that form a triangle.
Summary
Triangle inequalities guarantee that triangles are geometrically possible and determine relationships between sides and angles. Mastery of these inequalities supports later work in proofs, trigonometry, and coordinate geometry.