Turn in all binders, Math Whiz Punch Cards and HW paragraphs on How and Why do we create things? What are the consequences? Bell Ringer – Worksheet p. 30 Paragraph Proofs – Look pn p. 127 Make sure you have your project completed on FUNSIZE CANS. Take out your postulates sheet with the first page filled out with 7 postulates and 1 theorem.

Over Lesson 2–4 5-Minute Check 6 A.1 B.2 C.3 D.4 How many noncollinear points define a plane?

CCSS Content Standards G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Mathematical Practices 2 Reason abstractly and quantitatively. 3 Construct viable arguments and critique the reasoning of others.

Then/Now Identify and use basic postulates about points, lines, and planes. Write two-column proofs.

Vocabulary postulate axiom proof theorem deductive argument paragraph proof two-column proof informal proof

Concept

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 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. Identifying Postulates

Example 1 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. A. Plane P contains points E, B, and G.

Example 1 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. 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.

Example 2 Analyze Statements Using Postulates 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. 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.

Example 2 Analyze Statements Using Postulates Answer: Never; noncollinear points do not lie on the same line by definition. B. Determine whether the following statement is always, sometimes, or never true. Explain. contains three noncollinear points.

Do p , 3, 6, 7, 8, 10, 13, 14 HW p even Read 2-6 Take Notes (Copy all Properties on sheet) Missing Work Progress Report Signatures (some)

Concept

Example 3 Write a Paragraph Proof Given: Prove:ACD is a plane.

N StatementsReasons Proof: Two-column Proofs 1. Line AC intersects Line CD 1. Given 2. Line AC and Line CD must intersects at C 2. Postulate 2.6 – If two lines intersect, then their intersection is exactly one point. 3. Point A is on Line AC and Point D is on line CD. 4. Points A, C and D are non-collinear. 5. ACD is a plane 4. Postulate 2.4 – A plane contains at least three non-collinear points. 3. Postulate 2.3 – A line contains at least two points. 5. Postulate 2.2 – Through any three non-collinear points, there is exactly one plane.

Concept 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.