Lesson 5.7: Triangle Classification & Mixed Properties

Learning Objectives

• Classify triangles by sides and angles.
• Understand relationships between sides and angles.
• Apply mixed properties to solve triangle problems.

Topics Covered

• Triangle classification by sides: equilateral, isosceles, scalene
• Triangle classification by angles: acute, right, obtuse
• Side-angle relationships
• Triangle inequality theorem
• Mixed property applications

Key Terms

• Equilateral Triangle: All sides and angles are equal.
• Isosceles Triangle: Two sides and angles are equal.
• Scalene Triangle: No sides or angles are equal.
• Acute Triangle: All angles less than 90°.
• Right Triangle: One angle is exactly 90°.
• Obtuse Triangle: One angle greater than 90°.
• Triangle Inequality: Sum of any two sides is greater than the third side.

Lesson Examples

Example 1: Classifying Triangles by Sides

1. Triangle with sides 5, 5, 5 → Equilateral
2. Triangle with sides 6, 6, 8 → Isosceles
3. Triangle with sides 4, 5, 6 → Scalene

Example 2: Classifying Triangles by Angles

1. Triangle with angles 50°, 60°, 70° → Acute
2. Triangle with angles 30°, 60°, 90° → Right
3. Triangle with angles 40°, 50°, 100° → Obtuse

Example 3: Triangle Inequality

1. Check if sides 3, 4, 8 form a triangle → 3 + 4 < 8 → Not a triangle
2. Check sides 5, 7, 10 → 5 + 7 > 10, 5 + 10 > 7, 7 + 10 > 5 → Valid triangle

Practice Problems

1. Classify a triangle with sides 7, 7, 9 and angles 50°, 50°, 80°.
2. Determine if sides 2, 3, 6 can form a triangle.
3. Classify a triangle with angles 60°, 70°, 50° by type.

Tips & Common Mistakes

• Always check triangle inequality before classification.
• Remember: opposite sides are opposite their angles.
• Scalene triangles have all sides different; don’t confuse with isosceles.

Summary

Triangles can be classified by sides and angles. The triangle inequality and side-angle relationships are essential for understanding their properties and solving problems.

Challenge Problems

1. Given sides 8, 15, 17, classify the triangle by sides and angles.
2. Determine if a triangle with angles 45°, 45°, 90° is isosceles or scalene.
3. Explain why a triangle cannot have sides 1, 2, 4.