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3 2-5 Objectives You will learn to: Identify and use basic postulates about points, lines, and planes. Write paragraph proofs.

4 Vocabulary Postulate Theorem Proof Paragraph Proof Information Proof

5 Postulates In geometry, a postulate is a statement that describes a fundamental relationship between the basic terms of geometry. Postulates are accepted as true. 2.1 Through any two points, there is exactly one line. 2.2 Through any three points not on the same line, there is exactly one plane.

6 SNOW CRYSTALS Some snow crystals are shaped like regular hexagons. How many lines must be drawn to interconnect all vertices of a hexagonal snow crystal? Explore The snow crystal has six vertices since a regular hexagon has six vertices. Plan Draw a diagram of a hexagon to illustrate the solution.

Solve Label the vertices of the hexagon A, B, C, D, E, and F. Connect each point with every other point. Then, count the number of segments. Between every two points there is exactly one segment.

7 Solve Label the vertices of the hexagon A, B, C, D, E, and F. Connect each point with every other point. Then, count the number of segments. Between every two points there is exactly one segment. Be sure to include the sides of the hexagon. For the six points, fifteen segments can be drawn. Examine In the figure, are all segments that connect the vertices of the snow crystal. Answer: 15

8 ART Jodi is making a string art design. She has positioned ten nails, similar to the vertices of a decagon, onto a board. How many strings will she need to interconnect all vertices of the design? Answer: 45

9 Postulates 2.3 A line contains at least two points. 2.4 A plane contains at least three points not on the same line. 2.5 If two points lie in a plane, then the entire line containing those points lies in the plane. 2.6 If two lines intersect, then their intersection is exactly one point. 2.7 If two planes intersect, then their intersection is a line.

10 Determine whether the following statement is always, sometimes, or never true. Explain. If plane T contains plane T contains point G. contains point G, then 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.

11 Determine whether the following statement is always, sometimes, or never true. Explain. For, if X lies in plane Q and Y lies in plane R, then plane Q intersects plane R. Answer: Sometimes; planes Q and R can be parallel, and can intersect both planes.

12 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.

13 Determine whether each statement is always, sometimes, or never true. Explain. a. Plane A and plane B intersect in one point. Answer: Never; Postulate 2.7 states that if two planes intersect, then their intersection is a line. b. 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. Answer: Always; Postulate 2.1 states that through any two points, there is exactly one line. c. Two planes will always intersect a line. Answer: Sometimes; Postulate 2.7 states that if the two planes intersect, then their intersection is a line. It does not say what to expect if the planes do not intersect.

14 Proof A proof is an argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true. An important part of writing a proof is giving justifications to show that every step is valid.

15 Paragraph Proof A paragraph proof is a style of proof that presents the steps of the proof and their matching reasons as sentences in a paragraph. Although this style of proof is less formal than a two-column proof, you still must include every step. Five essential parts of a good proof: State the theorem or conjecture to be proven. List the given information. If possible, draw a diagram to illustrate the given information. State what is to be proved. Develop a system of deductive reasoning.

16 Given intersecting, write a paragraph proof to show that A, C, and D determine a plane. Given: intersects Prove: ACD is a plane. Proof: 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 Therefore, points A and D are not collinear. Therefore, ACD is a plane as it contains three points not on the same line.

17 Given is the midpoint of and X is the midpoint of write a paragraph proof to show that

Proof: We are given that S is the midpoint of and

definition of congruent segments, If then Since S

18 Proof: We are given that S is the midpoint of and X is the midpoint of By the definition of midpoint, Using the definition of congruent segments, statement Also using the given and the definition of congruent segments, If then Since S and X are midpoints, By substitution, and by definition of congruence,

19 What did you learn today? How to: Identify and use basic postulates about points, lines, and planes. Write paragraph proofs.

20 Assignment: Page odd, 23, 25, 27, 36

2-5 Postulates and Paragraph Proofs

2-5 Postulates and Paragraph Proofs Determine whether each statement is always, sometimes, or never true. Explain your reasoning. 7. The intersection of three planes is a line. If three planes intersect, then their intersection may be a