Mathematical Proofs: Direct and Indirect

Direct Proof

• A direct proof assumes a premise (p) is true, then uses properties, postulates, definitions, and theorems to show that the conclusion (q) is also true.
• Steps to write a direct proof:
  • State the given information.
  • State what needs to be proven.
  • Draw a figure (if applicable).
  • Present the proof (paragraph, two-column, or flow chart).

Indirect Proof

• An indirect proof assumes the conclusion is false, then uses properties, postulates, definitions, and theorems to show that the premise is also false, leading to a contradiction.
• Steps to write an indirect proof:
  • Accept the given statement as true.
  • Assume the opposite of the statement to be proved.
  • State the reason directly until there is a contradiction of the given or the assumed statement.
  • State the assumption is false.
  • Draw a figure (if applicable).
  • Present the proof (paragraph, two-column, or flow chart).

Properties of Equality

• Reflexive Property: Any quantity is equal to itself (a = a).
• Symmetric Property: If a = b, then b = a.
• Transitive Property: If a = b and b = c, then a = c.
• 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 a * c = b * c.
• Division Property: If a = b and c ≠ 0, then a/c = b/c.
• Substitution Property: If a = b, then b can be substituted for a in any expression or equation.

Example Problems/Proofs:

• Provided example problems illustrate how to apply direct and indirect proofs in geometry, including a proof that the midpoint of a line segment divides it into two congruent segments.

Activity Questions

• Questions are designed to test understanding of the different properties of equality.

Answer Key

• Correct answers to the activity questions are presented.