Lesson 14.4: Proof Strategies: Direct, Indirect, Contrapositive Mixed Exercises

Learning Objectives

• Understand and identify direct, indirect, and contrapositive proofs.
• Apply proof strategies to solve geometric problems.
• Recognize when each strategy is most appropriate.

Topics Covered

• Direct proof strategy
• Indirect (contradiction) proof strategy
• Contrapositive proof strategy
• Mixed exercise applications
• Logical reasoning in proofs

Key Terms

• Direct Proof: A proof that proceeds logically from given statements to the conclusion.
• Indirect Proof: A proof by contradiction, assuming the opposite of what you want to prove.
• Contrapositive Proof: Proving an implication by showing that the contrapositive is true.
• Implication: A logical statement of the form "If P, then Q."
• Contradiction: A logical inconsistency that shows an assumption is false.

Lesson Examples

Example 1: Direct Proof

Prove: The sum of two even numbers is even.

1. Let the numbers be 2a and 2b.
2. Their sum is 2a + 2b = 2(a + b).
3. Since 2(a + b) is divisible by 2, the sum is even.

Example 2: Indirect Proof

Prove: √2 is irrational.

1. Assume √2 is rational → √2 = p/q in lowest terms.
2. Then 2 = p²/q² → p² = 2q² → p² is even → p is even.
3. Let p = 2k → (2k)² = 4k² = 2q² → q² = 2k² → q is even.
4. Both p and q are even → contradicts lowest terms assumption → √2 is irrational.

Example 3: Contrapositive Proof

Prove: If n² is odd, then n is odd.

1. Contrapositive: If n is even, then n² is even.
2. Let n = 2k → n² = (2k)² = 4k² → n² is even.
3. Contrapositive true → original statement true.

Practice Problems

1. Prove by direct proof: The sum of two odd numbers is even.
2. Prove by indirect proof: There are infinitely many prime numbers.
3. Prove by contrapositive: If n² is divisible by 3, then n is divisible by 3.
4. Mixed: Determine which proof strategy is most effective for each given problem.

Tips & Common Mistakes

• Always clearly define your assumptions.
• For indirect proofs, explicitly state the contradiction.
• Check that the contrapositive is logically equivalent before using it.
• Don’t confuse direct reasoning with proving by example.

Summary

Direct, indirect, and contrapositive proofs are foundational proof strategies in geometry. Understanding when and how to apply each strengthens logical reasoning and problem-solving skills.

Challenge Problems

1. Use contrapositive proof to show: If the product of two integers is even, then at least one is even.
2. Prove by contradiction: There is no smallest positive rational number.
3. Determine which strategy to use to prove: The sum of the angles in any triangle is 180°.