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:  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: 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: 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:  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

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

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

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: 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. 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. Example 1

B. Points A and C 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. B. Points A and C determine a line. Answer: Example 1

B. Points A and C 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. 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. 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. 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

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

If plane T contains contains point G, then plane T contains point G. 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: Example 2

If plane T contains contains point G, then plane T contains point G. 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. Example 2

contains three noncollinear points. Analyze Statements Using Postulates B. Determine whether the following statement is always, sometimes, or never true. Explain. contains three noncollinear points. Answer: Example 2

contains three noncollinear points. 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. 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

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

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

Concept

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 Answer: Proof: 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. Example 4

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

Concept

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

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

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