Unit 1: Introduction to Proofs in Geometry

Initial Problems

• Attempt initial problems individually:
  • Solve for x: did you get x=5? Solutions will be reviewed.
  • Solve for y: did you get y=-1?
• Purpose: Practice solving and later review methods.

Introduction to Proofs

• Proofs: Why do we do what we do?
• Learning Objectives:
  • Learn different types of proofs.
  • Ability to identify and use appropriate proof types.
  • Determine the reason for the first statement of a proof.

Geometry as a New Language

• Geometry involves learning new vocabulary, like learning a new language.
• Practice is critical: can't just read, must engage actively.

Algebraic Properties of Equality

• Addition Property: If a = b, then a + c = b + c.
• Subtraction Property: If a = b, then a - c = b - c.
• Multiplication Property: If a = b, then ac = bc.
• Division Property: If a = b, and c ≠ 0, then a/c = b/c.
• Substitution Property: If a = b, then a can replace b in any expression.

Reflexive, Symmetric, and Transitive Properties

• Reflexive Property: a = a.
• Symmetric Property: If a = b, then b = a.
• Transitive Property: If a = b and b = c, then a = c.
• Application in solving equations and proofs.

Solving a Two-Step Equation

• Example: 3x + 4 = 19
  • Subtract 4: Subtraction Property of Equality
  • Divide by 3: Division Property of Equality

Solving More Complex Equations

• Example: 3(7y - 10) - 5y + 2y = 42y
• Steps:
  1. Use PEMDAS to simplify.
  2. Combine like terms.
  3. Apply Distributive Property.
  4. Use algebraic properties to isolate the variable.

Types of Proofs

• Two-column Proof: Statements and reasons in a tabular format.
• Paragraph Proof: Explanation in a written paragraph format.
• Flow Proof: Diagrams and arrows show the logical sequence, not used in this course.

Planning Proofs

• Always begin with the given statement.
• Use logical steps to reach the conclusion.
• Develop a plan of attack for solving proofs.

Example Proofs

• Two-column Proof: Segment bisectors and midpoints as examples to illustrate proof structure.
• Paragraph Proof: Converts a two-column proof into a narrative form.

Conclusion

• Summary of proof types and importance of logical reasoning.
• Encouragement to plan proofs and use learned information effectively.
• Reminder of available resources and ways to contact for help.