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Five-Minute Check (over Lesson 2–4) CCSS Then/Now New Vocabulary Postulates: Points, Lines, and Planes Key Concept: Intersections of Lines and Planes Example 1: Real-World Example: Identifying Postulates Example 2: Analyze Statements Using Postulates Key Concept: The Proof Process Example 3: Write a Paragraph Proof Theorem 2.1: Midpoint Theorem Bellringer 9/24/12 Define the following terms: • 1. Postulate • 2. Proof • 3. Theorem • 4. Algebraic Proof • 5. Two Column Proof • postulate • proof • theorem • algebraic proof • two column proof Identifying Postulates ARCHITECTURE Explain how the picture illustrates that the statement is true. Then state the postulate that can be used to show the statement is true. A. Points F and G lie in plane Q and on line m. Line m lies entirely in plane Q. Answer: Points F and G lie on line m, and the line lies in plane Q. Postulate 2.5, which states that if two points lie in a plane, the entire line containing the points lies in that plane, shows that this is true. Identifying Postulates ARCHITECTURE Explain how the picture illustrates that the statement is true. Then state the postulate that can be used to show the statement is true. B. Points A and C determine a line. Answer: Points A and C lie along an edge, the line that they determine. Postulate 2.1, which says through any two points there is exactly one line, shows that this is true. ARCHITECTURE Refer to the picture. State the postulate that can be used to show the statement is true. A. Plane P contains points E, B, and G. A. Through any two points there is exactly one line. B. A line contains at least two points. C. A plane contains at least three noncollinear points. D. A plane contains at least two noncollinear points. ARCHITECTURE Refer to the picture. State the postulate that can be used to show the statement is true. B. Line AB and line BC intersect at point B. A. Through any two points there is exactly one line. B. A line contains at least two points. C. If two lines intersect, then their intersection is exactly one point. D. If two planes intersect, then their intersection is a line. Analyze Statements Using Postulates A. Determine whether the following statement is always, sometimes, or never true. Explain. If plane T contains contains point G, then plane T contains point G. Answer: Always; Postulate 2.5 states that if two points lie in a plane, then the entire line containing those points lies in the plane. Analyze Statements Using Postulates B. Determine whether the following statement is always, sometimes, or never true. Explain. contains three noncollinear points. Answer: Never; noncollinear points do not lie on the same line by definition. A. Determine whether the statement is always, sometimes, or never true. Plane A and plane B intersect in exactly one point. A. always B. sometimes C. never B. Determine whether the statement is always, sometimes, or never true. Point N lies in plane X and point R lies in plane Z. You can draw only one line that contains both points N and R. A. always B. sometimes C. never Justify Each Step When Solving an Equation Solve -5(x+4) = 70. Algebraic Steps -5(x+4) = 70 Properties Original equation -5x + (-5)4 = 70 Distributive Property -5x - 20 = 70 Substitution Property -5x - 20 + 20 = 70 + 20 Addition Property Justify Each Step When Solving an Equation –5x = 90 -5x = 90 -5 -5 x = -18 Answer: a = –18 Substitution Property Division Property Substitution Property Justify Each Step When Solving an Equation Solve 2(5 – 3a) – 4(a + 7) = 92. Algebraic Steps 2(5 – 3a) – 4(a + 7) = 92 Properties Original equation 10 – 6a – 4a – 28 = 92 Distributive Property –18 – 10a = 92 Substitution Property –18 – 10a + 18 = 92 + 18 Addition Property Justify Each Step When Solving an Equation –10a = 110 Substitution Property Division Property a = –11 Answer: a = –11 Substitution Property Solve –3(a + 3) + 5(3 – a) = –50. A. a = 12 B. a = –37 C. a = –7 D. a = 7 Write an Algebraic Proof Begin by stating what is given and what you are to prove. Write an Algebraic Proof Proof: Statements Reasons 1. d = 20t + 5 1. Given 2. d – 5 = 20t 2. Addition Property of Equality 3. 4. =t 3. Division Property of Equality 4. Symmetric Property of Equality Which of the following statements would complete the proof of this conjecture? If the formula for the area of a trapezoid is , then the height h of the trapezoid is given by . Proof: Statements Reasons 1. 1. Given ? 2. _____________ 2. Multiplication Property of Equality 3. 3. Division Property of Equality 4. 4. Symmetric Property of Equality A. 2A = (b1 + b2)h B. C. D. Write a Geometric Proof If A B, mB = 2mC, and mC = 45, then mA = 90. Write a two-column proof to verify this conjecture. Write a Geometric Proof Proof: Statements Reasons 1. A B; mB = 2mC; mC = 45 1. Given 2. mA = mB 2. Definition of 3. mA = 2mC 3. Transitive Property of Equality 4. mA = 2(45) 4. Substitution 5. mA = 90 5. Substitution angles Proof: Statements 1. 2. Reasons 1. Given ? 2. _______________ 3. AB = RS 3. Definition of congruent segments 4. AB = 12 4. Given 5. RS = 12 5. Substitution A. Reflexive Property of Equality B. Symmetric Property of Equality C. Transitive Property of Equality D. Substitution Property of Equality