2-1 Study Guide and Intervention | Study Guide

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High School
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High School Algebra
Academic year:

2023
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NAME ____________________________________________ DATE ____________________________ PERIOD _____________ 2-1 Study Guide and Intervention Conjectures and Counterexamples Make Conjectures Inductive reasoning is reasoning that uses information from different examples to form a conclusion or statement called a conjecture. Example 1: Write a conjecture about the next Example 2: Write number in the sequence 1, 3, 9, 27, 81. Look for a pattern: Each number is a power of 3. 1 3 9 27 81 ! " # $ 3 3 3 3 3% Conjecture: Starting with 1, to get the next term, multiply the previous term by 3. The next number will be 3& or 243. a conjecture about the number of small squares in the next figure. Look for a pattern: The sides of the squares have measures 1, 2, and 3 units. Conjecture: Starting with , to get the next term, increase each side of the square by 1 unit from the previous term. For the next figure, the side of the square will be 4 units, so the figure will have 16 small squares. Exercises Write a conjecture that describes the pattern in each sequence. Then use your conjecture to find the next item in the sequence. 1. –5, 10, –20, 40 2. 1, 10, 100, 1000 ' ( ) 3. 1, &, &, & Write a conjecture about each value or geometric relationship. 4. A(–1, –1), B(2, 2), C(4, 4) 6. ∠ABC and ∠DBE are vertical angles. Chapter 2 5. ∠1 and ∠2 form a right angle. 7. ∠E and ∠F are right angles. 5 Glencoe Geometry NAME ____________________________________________ DATE ____________________________ PERIOD _____________ 2-1 Study Guide and Intervention (continued) Conjectures and Counterexamples Find Counterexamples A conjecture is false if there is even one situation in which the conjecture is not true. The false example is called a counterexample. Example: Find a counterexample to show the conjecture is false. %%%% ≅ %%%% %%%%. If 𝐴𝐵 𝐵𝐶 , then B is the midpoint of 𝐴𝐶 %%%% ≅ %%%% Is it possible to draw a diagram with 𝐴𝐵 𝐵𝐶 such that B is not the midpoint? This diagram is a counterexample because point B is not on %%%% 𝐴𝐶 . The conjecture is false. Exercises Determine whether each conjecture is true or false. Give a counterexample for any false conjecture. 1. If points A, B, and C are collinear, then AB + BC = AC. 2. If ∠R and ∠S are supplementary, and ∠R and ∠T are supplementary, then ∠T and ∠S are congruent. 3. If ∠ABC and ∠DEF are supplementary, then ∠ABC and ∠DEF form a linear pair. %%%% ⊥ %%%% 4. If 𝐷𝐸 𝐸𝐹 , then ∠DEF is a right angle. Chapter 2 6 Glencoe Geometry