# Direct and Indirect Proofs

Activity 1: Fact of Fake?

1. Fake

2. Fake

3. True

4. True

5. True

6. Fake

7. Fact

8. Fact

Guide a

Questions:

1. I identified each of number by just carefully observing them.

2. Because Identifying each of them is not as difficult as other activities.

3. Because knowing if it is either a fact or fake

4. I have realized that this type of information is very important in my decision making in the future.

Activity 2: Am I Filipino?

Statements | Reasons |

1. I am a Capizeno | 1. Given |

2. I was born in Capiz | 2. Place of birth |

3. Capiz is one of the provinces in the Philippines. | 3. One of the Provinces of the Country |

4. My parents are both Filipino. | 4. Definition of Natural Born Citizen |

5. Therefore, I am a Filipino Citizen | 5. Statements 2, 4 and Definition of Natural Born Citizen. |

Guide Questions:

1. I am a Capizeño is the hypothesis of the activity

2. I am a Filipino is the conclusion of the activity

3. Because I was born in the Philippines and was raised by Filipino parents, and because I speak my native tongue

4.My proof is that, English is my second language and I go to a Filipino school.

5. By stating my thoughts on the topic.

6. I can try to explain the concepts related to the topic and differentiate each of them to prove it.

Activity 3: Know Me Well

A

1. Start with the given information.

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 (two-column or paragraph).

B.

1. Accept the given statement is true.

2. Assume the opposite of the statement to be proved.

3. State the reasons directly until there 1s a contrad1ct1on of the given or the other statements.

4. State that the assumption of the opposite of the statement to be proved must be false.

5. Follow Steps 3 and 4 of writing direct proofs.

Activity 4: Congruency as Its Best

A. Given that point B is the midpoint of line segment AC, by the definition of a midpoint.

It follows that the measurement of AB and AC are equal by the definition of midpoint. AB = BC by definition of congruent segments.

Thus. We can say that line segment AB is congruent to line segment BC, or AB ≅BC according to the definition of congruent segments.

B.

Statement | Reason |

1. B is the midpoint of AC | 1. Given. Definition of congruent segments |

2. AB=BC | 2. Definition of congruent segments |

3.AB=BC | 3. Definition of Midpoints |

4. AB ≅ BC | 4. Statements 2 and 3 |

Guide Questions:

1. The premise of the activity is when you had to state the proofs to complete the table.

2. To give reason for each of the given statements.

3. My second statements is AB=BC and its reason is because the other line segment will change to remain congruent with it.

4. They have the same congruent line segments.

5. Indirect proof

Activity 5: Contradiction to Realization

Statement | Reason |

1. ∠A. and ∠B are supplementary angles. | 1. Given |

2. ∠B ≥ 180° | 2. By Assumption |

3. ∠A + ∠B = 180° | 3. Definition of supplementary angles. |

4. Therefore, ∠B < 180. | 4. Contradiction in statements 2 and 3 |

Guide Questions:

1. The first step of indirect proof is instead of showing that the conclusion to be proved is true, you show that all of the alternatives are false.

2. proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion

3. The angle B is less than 180 degrees because when you look at from the given angles such as angle A and angle B are supplementary angles Therefore each angle measures less than supplementary which means angle A plus angle B is equal 180 degrees by the definition of supplementary angles.

Reflections in Guide Questions:

1. I have learned about writing both direct and indirect proofs.

2. It could apply through critical thinking reasoning the way we deliver our thoughts because an indirect proof relies on a contradiction to prove a given conjecture by assuming the conjecture is not true, and then running into a contradiction proving that the conjecture must be true.

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