Unit Four: Classifying Triangles

Introduction

• Moving on from points, lines, and planes to triangles.
• A triangle is formed by connecting three lines.
• "Tri" means three, hence three angles and three sides.

Classification by Angles

• Acute Triangle: All three angles are acute (less than 90 degrees).
• Obtuse Triangle: One angle is obtuse (greater than 90 degrees).
• Right Triangle: One angle is a right angle (90 degrees).
• Equiangular Triangle: All angles are congruent (equal in measure).
  • Equiangular implies all angles are equal.

Classification by Sides

• Equilateral Triangle
  • All sides are equal.
  • Equilateral triangles are also equiangular.
• Isosceles Triangle
  • Two sides are equal.
  • Angles opposite the equal sides are also equal.
• Scalene Triangle
  • No sides are equal.
  • All angles are different.

Using Diagrams to Classify Triangles

• Identify triangles by noting angle types or congruent sides.
• Use symbols for quick notation (e.g., Δ for triangle).

Example Classifications

• Right Triangle: Look for the box symbol indicating a 90-degree angle.
• Obtuse Triangle: Identify the largest angle more than 90 degrees.
• Acute Triangle: All angles are less than 90 degrees.
• Equilateral/Equiangular Triangle: All sides or all angles are marked equal.

Solving Problems with Triangles

• Use the relationship between sides and angles to solve problems.
• Bisecting Lines: If a line bisects an angle or a side, it divides it into equal parts.

Practical Examples

• Classify by Angles and Sides

  • Determine if triangles are right, obtuse, or acute based on angles.
  • Determine if triangles are equilateral, isosceles, or scalene based on sides.

• Algebra and Triangles

  • Use given conditions (e.g., equilateral, isosceles) to solve for unknowns.
  • Equations can be formed based on congruent sides or angles.
  • Solve equations to find missing lengths or angles.

Key Points

• Relationships between sides and angles are crucial in classifying triangles.
• Notations and symbols can simplify problem-solving.
• Algebra can be integrated to find unknown measures in geometric problems.
• Practice classifying triangles by both sides and angles to master the concept.