Mathematical Proofs: Direct and Indirect

Study Notes

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).

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).

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.

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.