Direct and Indirect Proof PDF

# Direct and Indirect Proof ## Direct Proof Suppose you are given a premise *p* and you want to prove that a conclusion *q* is true. The direct proof would assume that *p* is true, then use in the context of geometry, properties, postulates, definitions, and theorems to show *q* is true. ### Writ...

# Direct and Indirect Proof ## Direct Proof Suppose you are given a premise *p* and you want to prove that a conclusion *q* is true. The direct proof would assume that *p* is true, then use in the context of geometry, properties, postulates, definitions, and theorems to show *q* is true. ### Writing Direct Proof 1. State the given. These statements are considered facts, therefore true. 2. State what to prove. 3. Draw a figure, which can serve as a guide in establishing the proof. 4. Present the proof using a preferred method. (Paragraph Form, Two-column Form, Flow Chart) ### Example 1 Given: M is the midpoint of AB. Prove: AM = BM. <p align="center"> <img src="https://i.imgur.com/4t86S2N.png" alt="A line segment AB with point M in the middle."> </p> #### Using Paragraph Form Given that M is the midpoint of line segment AB, then AM = BM by the definition of a midpoint. Since AM = BM, then by the definition of congruent segments, we can say that line segment AM is congruent to line segment BM or AM = BM. #### Using Two-Column Form Given: M is the midpoint of AB. Prove: AM = BM. <p align="center"> <img src="https://i.imgur.com/4t86S2N.png" alt="A line segment AB with point M in the middle."> </p> | Statement | Reason | | -------------------------------------------------------- | ----------------------------------- | | M is the midpoint of AB | Given | | AM = BM | Definition of a Midpoint | | AM = BM | Definition of Congruent Segment | #### Using Flow Chart Given: M is the midpoint of AB. Prove: AM = BM. <p align="center"> <img src="https://i.imgur.com/4t86S2N.png" alt="A line segment AB with point M in the middle."> </p> <p align="center"> <img src="https://i.imgur.com/kXwQmN2.png" alt="FlowChart representing direct proof, a diagram showing the steps of the proof."> </p> ## Indirect Proof Given a premise *p* and a conclusion *q,* an indirect proof would assume that *q* is false. You would then use the same properties, postulates, definitions, and theorems to show that *p* would also be false by arriving at a contradiction. ### Writing Indirect Proof 1. Accept the given statement is true. 2. Assume the opposite of the statement to be proved. 3. State the reason directly until there is a contradiction of the given or the other statement. 4. State that the assumption of the opposite statement to be proved must be false. 5. Draw a figure, which can serve as a guide in establishing the proof. 6. Present the proof using a preferred method. (Paragraph Form, Two-column Form, Flow Chart) ### Example 1 Given: M is not the midpoint of AB. Prove: AM ≠ BM. <p align="center"> <img src="https://i.imgur.com/Q2X44O0.png" alt="A line segment AB with point M in the middle, M is not the midpoint of AB."> </p> #### Using Paragraph Form Assume temporarily that AM = BM, then M is the midpoint of AB by the definition of a midpoint. However, it contradicts the given; therefore, AM ≠ BM. #### Using Two-Column Form Given: M is not the midpoint of AB. Prove: AM ≠ BM. <p align="center"> <img src="https://i.imgur.com/Q2X44O0.png" alt="A line segment AB with point M in the middle, M is not the midpoint of AB."> </p> | Statement | Reason | | -------------------------------------------------------- | ----------------------------------- | | M is not the midpoint of AB | Given | | AM = BM | Assumption | | M is the midpoint of AB | Definition of a midpoint | | AM + BM | Contradiction in the statement | #### Using Flow Chart Given: M is not the midpoint of AB Prove: AM ≠ BM <p align="center"> <img src="https://i.imgur.com/Q2X44O0.png" alt="A line segment AB with point M in the middle, M is not the midpoint of AB."> </p> <start_of_image>p align="center"> <img src="https://i.imgur.com/U34qT44.png" alt="Flow Chart representing indirect proof, a diagram showing the steps of the proof."> </p> ## Activity 1. Which property of equality states that a quantity is always equal to itself? - **Symmetric Property** - **Reflexive Property** - Transitive Property - Addition Property 2. If *x* = 5, Which property of equality justifies that 5 = *x*? - Transitive Property - **Symmetric Property** - Reflexive Property - Substitution Property 3. If *a* = *b* and *b* = *c*, what property of equality allows us to conclude that *a* = *c*? - **Transitive Property** - Symmetric Property - Reflexive Property - Addition Property 4. If *x* = 7, what property of equality justifies replacing *x* with 7 in the equation *x* + *y* =12? - Reflexive Property - Subtraction Property - **Substitution Property** - Multiplication Property 5. Which property of equality is used when we conclude that *a* + *c* = *b* + *c* from *a* = *b*? - **Addition Property** - Division Property - Reflexive Property - Transitive Property 6. Given *m* = *n* and *c* = 2, which property of equality justifies that *mc* = *nc*? - Addition Property - Symmetric Property - **Multiplication Property** - Division Property 7. What property of equality allows you to conclude that *a* - *d* = *b* - *d* if *a* = *b*? - Symmetric Property - Addition Property - **Subtraction Property** - Division Property 8. Which property of equality allows you to divide both sides of 6*p* = 18 by 6 to get *p* = 3? - Substitution Property - **Division Property** - Multiplication Property - Addition Property 9. If *x* = *y*, which property of equality allows you to substitute *y* for *x* in the equation *x* + *z* = 10? - Reflexive Property - Symmetric Property - **Substitution Property** - Transitive Property 10. Which property of equality justifies that 4 = 4? - Symmetric Property - **Reflexive Property** - Transitive Property - Subtraction Property ## Answer Key 1. b) Reflexive Property 2. b) Symmetric Property 3. a) Transitive Property 4. c) Substitution Property 5. a) Addition Property 6. c) Multiplication Property 7. c) Subtraction Property 8. b) Division Property 9. c) Substitution Property 10. b) Reflexive Property ## Two-Column Proof ### Problem: Given: 3*x* + 5 = 20 Prove: *x* = 5 | Statement | Reason | | -------------------------------------------------------- | ------------------- | | 1. 3*x* + 5 = 20 | Given | | 2. 3*x* = 20 - 5 | Simplification | | 3. 3*x* = 15 | Simplification | | 4. *x* = 15 / 3 | Simplification | | 5. *x* = 5 | Simplification |