Five-Minute Check (over Lesson 2–3) Mathematical Practices 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 an Algebraic Flow Proof Theorem 2.1: Midpoint Theorem Example 4: Write a Geometric Flow Proof Key Concept: How to Write a Paragraph Proof Example 5: Write a Paragraph Proof Lesson Menu

Determine whether the stated conclusion is valid based on the given information. If not, choose invalid. Given: A and B are supplementary. Conclusion: mA + mB = 180 A. valid B. invalid 5-Minute Check 1

Determine whether the stated conclusion is valid based on the given information. If not, choose invalid. Given: Polygon RSTU has 4 sides. Conclusion: Polygon RSTU is a square. A. valid B. invalid 5-Minute Check 2

Determine whether the stated conclusion is valid based on the given information. If not, choose invalid. Given: A and B are congruent. Conclusion: ΔABC exists. A. valid B. invalid 5-Minute Check 3

Determine whether the stated conclusion is valid based on the given information. If not, choose invalid. Given: A and B are congruent. Conclusion: A and B are vertical angles. A. valid B. invalid 5-Minute Check 4

Determine whether the stated conclusion is valid based on the given information. If not, choose invalid. Given: mY in ΔWXY = 90. Conclusion: ΔWXY is a right triangle. A. valid B. invalid 5-Minute Check 5

How many noncollinear points define a plane? B. 2 C. 3 D. 4 5-Minute Check 6

Mathematical Practices 2 Reason abstractly and quantitatively. 3 Construct viable arguments and critique reasoning of others. Content Standards G.CO.9 Prove theorems about lines and angles. G.MG.3 Apply geometric methods to solve problems. MP

You used deductive reasoning to prove statements. Analyze figures to identify and use postulates about points, lines, and planes. Analyze and construct viable arguments in several proof formats. Then/Now

postulate axiom proof deductive argument flow proof paragraph proof Vocabulary

Concept

Concept

A. Points F and G determine a line. 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 determine a line. Postulate 2.1, which says through any two points there is exactly one line. Example 1

Identifying Postulates Points F and G lie along an edge, the line that they determine. Postulate 2.1 shows that this is true. Answer: Points F and G lie along an edge, the line they determine. Postulate 2.1 states that through any two points, there is exactly one line. Example 1

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 D lie in plane BEC and on line n. Line n lies entirely in plane BEC. Postulate 2.5, states that if two points lie in a plane, the entire line containing the points lies in that plane. Example 1

Identifying Postulates Points A and D lie on line n, and the line lies in plane BEC. Postulate 2.5 shows that this is true. Answer: Points A and D lie on line n, and the line lies in plane BEC. Postulate 2.5 states that if two points lie in a plane, then the entire line containing the points lies in the plane. Example 1

ARCHITECTURE Refer to the picture 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. Example 1

ARCHITECTURE Refer to the picture 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. Example 1

The intersection of three planes is a line Analyze Statements Using Postulates A. Determine whether the following statement is always, sometimes, or never true. Explain your reasoning. The intersection of three planes is a line Postulate 2.7, states that if two planes intersect then their intersection is exactly one line. Visualize adding a third plane. What will the intersection look like? Example 2

So, sometimes the intersection of three planes will be a line. Analyze Statements Using Postulates The three plane could intersect in a line or the three planes could intersect at a point. So, sometimes the intersection of three planes will be a line. Answer: Sometimes; if three planes intersect, then their intersection could be a line or a point. Example 2

Through points H and K, there is exactly one line. Analyze Statements Using Postulates B. Determine whether the following statement is always, sometimes, or never true. Explain your reasoning. Through points H and K, there is exactly one line. Postulate 2.1, states through any two points there is exactly one line. So, always two points will be a line. Answer: Always; Postulate 2.1 states that through any two points, there is exactly one line. Example 2

A. Determine whether the statement is always, sometimes, or never true 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 Example 2

B. Determine whether the statement is always, sometimes, or never true 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 Example 2

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Prove that if 2(x + 7) = 6, then x = −4. Write a flow proof. Write an Algebraic Flow Proof Prove that if 2(x + 7) = 6, then x = −4. Write a flow proof. Given: 2(x + 7) = 6 Prove: x = −4 Answer: Proof: Given Division Property Simplify Subtraction Property Example 3

Concept

Given that , write a flow proof to show that x = 7. Write a Geometric Flow Proof Given that , write a flow proof to show that x = 7. Given: Prove: x = 7. Answer: Proof: Example 4

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Given: Prove: ACD is a plane. Write a Paragraph Proof Given: Prove: ACD is a plane. Proof: and must intersect at C because if two lines intersect, then their intersection is exactly one point. Point A is on and point D is on . Points A, C, and D are not collinear. Therefore, ACD is a plane as it contains three points not on the same line. Example 5

Example 5

Proof: ? Example 5

B. Segment Addition Postulate A. Midpoint Theorem B. Segment Addition Postulate C. Definition of congruent segments D. Substitution Example 5