Lecture Notes: Understanding Angles

Objectives

• Measure and classify angles
• Identify and use congruent angles and the bisector of an angle
• Apply the angle addition postulate

Introduction to Angles

• Example of real-life usage: Carpentry (miter joint)
• Importance of accurate angle measurement

Key Vocabulary

• Angle: Formed by two rays with a common endpoint (vertex)
  • Example: angle ABC, B is the vertex
• Sides of an angle: Rays AB and BC
• Naming Angles: Use three letters with the vertex in the middle (e.g., angle ABC)

Types of Angles

• Acute angle: Less than 90 degrees
• Obtuse angle: Greater than 90 degrees
• Right angle: Exactly 90 degrees, indicated by a square
• Straight angle: Exactly 180 degrees, a straight line

Examples and Naming

• Example 1:
  • Vertex: K
  • Sides: Ray KJ and Ray KL
  • Naming: Angle K, JKL, LJK (K is the vertex)
  • Classification: Obtuse (greater than 90 degrees)
• Example 2:
  • Vertex: S
  • Sides: Ray SR and Ray ST
  • Naming: Angle S, RST, TSR
  • Classification: Right angle (90 degrees)

Congruent Angles

• Definition: Angles with equal measures are congruent
• Notation: ≅ (congruent)
• Example: Angle A = 75 degrees, Angle B = 75 degrees → A ≅ B

Angle Bisector

• Definition: A ray dividing an angle into two congruent angles
• Example: Ray BD is a bisector; ABD ≅ DBC

Perpendicular Lines and Bisectors

• Perpendicular lines: Intersect at a right angle (symbol: ⊥)
• Perpendicular bisector: Perpendicular at a segment's midpoint
  • Example: Line LM is a perpendicular bisector of PQ

Examples of Naming and Classification

• Multiple ways to name angles
• Example 3:
  • Name CBF as FBC
  • Sides of EBD: Ray EB, Ray BD
  • ABC is a straight angle
  • Example of obtuse angles: ABF, EBD

Angle Addition Postulate

• Postulate: If D is in the interior of angle ABC, then ABD + DBC = ABC

Examples Using the Angle Addition Postulate

• Example 1: ABD = 48°, DBC = 78° → ABC = 126°
• Example 2: DBC = 74°, ABC = 119° → ABD = 45°

Solving Algebraic Angle Problems

• Example 3: Given full angle and expressions for segments, solve for x and individual angles
• Example 4: Use addition postulate to solve for unknown angles and algebraic expressions

Final Problem Review

• Solve for x and angle measurements using given expressions
• Example 9: Solve using 6x + 26 + 2x - 9 = 11x - 31, find x and angle ABF

Conclusion

• Encouragement to ask questions and seek clarification
• Follow-up tasks provided for further practice