2.5: Conjectures That Lead to Theorems

1. 2.5: Conjectures That Lead to Theorems Expectations: G1.1.1: Solve multistep problems and construct proofs involving vertical angles, linear pairs of angles, supplementary angles, complementary angles and right angles. G3.1.3: Find the image of a figure under the composition of two or more isometries and determine whether the resulting figure is a reflection, rotation, translation, or glide reflection image of the original figure. L3.1.1: Distinguish between inductive and deductive reasoning, identifying and providing examples of each. L3.1.2: Differentiate between statistical arguments (statements verified empirically using examples or data) and logical arguments based on the rules of logic. L3.3.1: Know the basic structure for the proof of an “if…, then” statement and that proving the contrapositive is equivalent. 2.5: Conjectures that Lead to Theorems

2. Inductive Reasoning Inductive reasoning is based on observations or patterns. Inductive reasoning is NOT valid for proofs. 2.5: Conjectures that Lead to Theorems

3. Inductive Reasoning Betty observed 5 white cars traveling very slowly down the road so she concludes that all white cars are slow. Betty used inductive reasoning to reach her conclusion. 2.5: Conjectures that Lead to Theorems

4. Deductive Reasoning Deductive reasoning is based on known facts, or statements such as postulates definitions or theorems. Deductive reasoning is valid for proofs. 2.5: Conjectures that Lead to Theorems

5. Deductive Reasoning Triangle ABC is a right triangle so Billy concludes triangle ABC has a right angle. Billy has used deductive reasoning because the definition of a right triangle tells him that it has a right angle. 2.5: Conjectures that Lead to Theorems

6. Which of the following statements demonstrates deductive reasoning? A. All crows are black, and all crows are birds; therefore, all birds are black. B. All dolphins have fins, and all fish have fins; therefore, all dolphins are fish. C. Edward is a human being, and all human beings are mortal; therefore, Edward is mortal. D. Megan gets good grades, and studying results in good grades; therefore, Megan is studying. 2.5: Conjectures that Lead to Theorems

7. Statistical Arguments vs Logical Arguments • A statistical argument is made using _____ or ___________ to justify your statements. • A logical argument is made by combining true statements (postulates, definitions and theorems) together to reach a conclusion. 2.5: Conjectures that Lead to Theorems

8. Is this a statistical argument or a logical argument? Barney read in his owners manual that he can increase his gas mileage by 15% if he slows down by an average of 10 miles per hour on the expressway. He then suggests to his brother that he too should slow down to save gas. 2.5: Conjectures that Lead to Theorems

9. Is Pebbles using a statistical argument or a logical argument? Given A, B and C are collinear points and that A and B are both on plane Q, Betty is trying to determine if C must also be on Q. Pebbles says C must be on Q because of the Unique Plane Postulate. 2.5: Conjectures that Lead to Theorems

10. Who is using a statistical argument? • Harry picked 3 cards out of a deck of cards and selected 3 hearts. • Ron observed this and said the deck must not be a regular deck of cards because there is no way you can draw 3 straight hearts from a deck of cards. • Hermione said it could be a regular deck because there is about a 1% chance of drawing 3 in a row of any suit. • only Ron • only Hermione • only Harry • Ron and Hermione • no one is using a statistical argument 2.5: Conjectures that Lead to Theorems

11. Congruent Supplements Theorem: If 2 angles are supplements of congruent angles (or the same angle), then they are __________. 2.5: Conjectures that Lead to Theorems

12. Congruent Supplements Theorem: 1 2 3 4 If ∠1 ≅ ∠ 3, ∠ 1 is supplementary ∠ 2 and ∠ 3 is supplementary ∠ 4, then ____________. 2.5: Conjectures that Lead to Theorems

13. 1 4 2 3 Vertical Angles Defn: Two angles are vertical angles iff they are the nonadjacent angles formed by two intersecting lines. 2.5: Conjectures that Lead to Theorems

14. 1 4 2 3 Vertical Angles Make a conjecture about vertical angles. Try to justify your conjecture with mathematical statements. 2.5: Conjectures that Lead to Theorems

15. 2.5: Conjectures that Lead to Theorems

16. Vertical Angle Theorem 2.5: Conjectures that Lead to Theorems

17. Prove the Vertical Angle Theorem 2.5: Conjectures that Lead to Theorems

18. Determine the measure of ∠1 1 x2 + 2x +4 2x2 + 5x -50 3 2.5: Conjectures that Lead to Theorems

19. Complete Activity 2 on page 119. You may work in pairs. You have 10 minutes. 2.5: Conjectures that Lead to Theorems

20. Two Reflections Theorem for Translations. If a transformation is the composite of two reflections over parallel lines, then it is 2.5: Conjectures that Lead to Theorems

21. You want to translate ΔABC 10 units by reflecting it twice. Describe as accurately as you can how to position the lines. 2.5: Conjectures that Lead to Theorems

22. Two Reflections Theorem For Rotations If a transformation is the composite of two reflections over intersecting lines, then it is 2.5: Conjectures that Lead to Theorems

23. l 70° m F Two Reflections Theorem for Rotations F” F’ 2.5: Conjectures that Lead to Theorems

24. Two Reflections Theorem for Rotations l F” 70° m 140° F 2.5: Conjectures that Lead to Theorems

25. Point P is reflected over line m and then over line n. If the overall result is a rotation of 80 degrees, what is the measure of the acute angle formed by lines m and n? • 20 • 40 • 80 • 140 • 160 2.5: Conjectures that Lead to Theorems

26. Assignment • pages 121 – 125, • # 10 – 24 (evens), 25 – 27, 30, 32 – 34, 36, 38, 47, 48, 50 2.5: Conjectures that Lead to Theorems