DIRECT-AND-INDIRECT-PROOF

DIRECT AND INDIRECT PROOF

Definition of Proof

  • A logical argument where each statement is supported by an accepted true statement.

Types of Proof

  • Direct Proof: A straightforward logical argument.

  • Indirect Proof (Proof by Contradiction): Assumes the opposite of what is to be proved and derives a contradiction.

  • Informal Proof: Less formal structure, using logical reasoning.

  • Formal Proof: Rigid structure following established rules.

  • Paragraph Form: Written in narrative style explaining steps.

  • Two-Column Form: Lists statements and corresponding reasons in two separate columns.

  • Flowchart: Visual representation of the steps in the proof.

Writing Proof - Steps to Follow

Step 1: Understand the Theorem

  • Read and interpret the theorem thoroughly.

Step 2: Draw a Figure

  • Create an accurate figure and label all parts appropriately.

Step 3: State Hypothesis and Conclusion

  • Clearly identify the hypothesis as "GIVEN" and the conclusion as "PROVE."

Step 4: Write the Proof

  • Compose the proof detailing both statements and reasons supporting each step.

Example of Informal Proof in Paragraph Form

Proving Complements of Congruent Angles Are Congruent

  1. Read and Understand the Theorem: Ensure comprehension of the theorem that states if angles are the complements of congruent angles, then those angles are also congruent.

  2. Identify involved angles and their relationships in a drawn figure.

  3. State the relationships, clearly adding that if Angle B is congruent to Angle D, and Angle A is a complement of Angle B, while Angle C is a complement of Angle D, then prove that Angle A is congruent to Angle C.

Detailed Steps to Complete Proof

  • Step 1: Restate the theorem.

  • Step 2: Draw and label the figure accurately showing congruencies between angles.

  • Steps with Hypothesis:

    • Given: Angle B ≅ Angle D.

    • Angle A is a complement of Angle B.

    • Angle C is a complement of Angle D.

    • Prove: Angle A ≅ Angle C.

    • State that since the sum of the measures of complementary angles is always 90°, use the Transitive Property of Equality to show relations between angles.

Fill in the missing details for the proof

  • The sum of complementary angles is (a) 90°.

  • By property of equality, Angle A + Angle B = Angle C + Angle D.

  • Angles B and D are congruent; therefore, their measures are equal (b).

  • Substitute measures for equality (c).

  • Eliminate B from the equation using the Subtraction Property, resulting in measurement of Angle A = measurement of Angle C (d).

Form of Proofs

Two-Column Form

  • Statement and Reason format:

    1. Given: Lines intersecting at point O forming angles 1 and 2 as vertical angles.

    2. Apply the Definition of Linear Pair to show the properties of the angles involved.

Flowchart Method

  • To Prove: The sum of the angles in a triangle equals (TIAT) 180°.