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2.5 Postulates & Paragraph Proofs Objectives Identify and use basic postulates about points, lines, and planes. Write paragraph proofs. Postulates In geometry, a postulate is a statement that describes a fundamental relationship between the basic terms of geometry. Postulates are always accepted as true. Postulates Postulate 2.1 – Through any two points, there is exactly one line. Postulate 2.2 – Through any three points not on the same line, there is exactly one plane. Example 1: 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. Example 1: 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. Example 1: Examine In the figure, are all segments that connect the vertices of the snow crystal. Answer: 15 Your Turn: 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 More Postulates Postulate 2.3 – A line contains at least two points. Postulate 2.4 – A plane contains at least three points not on the same line. Postulate 2.5 – If two points lie in a plane, then the entire line containing those points lies in that plane. And Even More Postulates Postulate 2.6 – If two lines intersect, then their intersection is exactly one point. Postulate 2.7 – If two planes intersect, then their intersection is a line. Example 2a: 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. Example 2b: 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. Example 2c: 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. Your Turn: 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. Your Turn: Determine whether each statement is always, sometimes, or never true. Explain. 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. Theorems We use undefined terms, definitions, postulates, and algebraic properties of equality to prove that other statements or conjectures are true. Once a statement or conjecture has been shown to be true, it is called a theorem. Once proven true, a theorem can be used like a definition or postulate to justify other statements or conjectures. Paragraph Proofs Proof – a logical argument in which each statement you make is supported by a statement that is accepted as true Paragraph Proof – a type of proof in which you write a paragraph to explain why a conjecture for a given situation is true (also, referred to as an “informal proof ”) Paragraph Proofs There are 5 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. Paragraph Proofs Hint: Before writing a proof, you should have a plan. One strategy is to work backwards. Start with what you want to prove, and work backwards step by step until you reach the given information. Theorems Theorem 2.8 (Midpoint Theorem) If M is the midpoint of AB, then AM MB. Example 3: 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. Your Turn: Given midpoint of is the midpoint of and X is the write a paragraph proof to show that Your Turn: Proof: We are given that S is the midpoint of X is the midpoint of and By the definition of midpoint, Using the definition of congruent segments, Also using the given statement and the definition of congruent segments, If then Since S and X are midpoints, By substitution, congruence, and by definition of Assignment Geometry: Pg. 92 #12 - 27 Pre-AP Geometry: Pg. 92 # 12 - 28