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NAME DATE 2-5 PERIOD Study Guide and Intervention Postulates and Paragraph Proofs Points, Lines, and Planes In geometry, a postulate is a statement that is accepted as true. Postulates describe fundamental relationships in geometry. Postulate Postulate Postulate Postulate Postulate Postulate Postulate 2.1: 2.2: 2.3: 2.4: 2.5: 2.6: 2.7: Example Through any two points, there is exactly one line. Through any three noncollinear points, there is exactly one plane. A line contains at least two points. A plane contains at least three noncollinear points. If two points lie in a plane, then the entire line containing those points lies in the plane. If two lines intersect, then their intersection is exactly one point. If two planes intersect, then their intersection is a line. Determine whether each statement is always, sometimes, or never true. a. There is exactly one plane that contains points A, B, and C. Sometimes; if A, B, and C are collinear, they are contained in many planes. If they are noncollinear, then they are contained in exactly one plane. b. Points E and F are contained in exactly one line. Always; the first postulate states that there is exactly one line through any two points. Exercises Determine whether each statement is always, sometimes, or never true. 1. A line contains exactly one point. 2. Noncollinear points R, S, and T are contained in exactly one plane. 3. Any two lines ℓ and m intersect. −−− 4. If points G and H are contained in plane M, then GH is perpendicular to plane 5. Planes M. R and S intersect in point T. −− −−− −− 6. If points A, B, and C are noncollinear, then segments AB, BC, and CA are contained in exactly one plane. −− −− −− −− In the figure, AC and DE are in plane Q and AC || DE. State the postulate that can be used to show each F statement is true. Q C 7. Exactly one plane contains points F, B, and E. B A D "#$ lies in plane 8. BE Chapter 2 005_054_GEOCRMC02_890511.indd 31 E Lesson 2-5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. c. Two lines intersect in two distinct points M and N. Never; the intersection of two lines is one point. G Q. 31 Glencoe Geometry PDF Pass 4/10/08 2:13:55 PM NAME 2-5 DATE Study Guide and Intervention PERIOD (continued) Postulates and Paragraph Proofs Paragraph Proofs A logical argument that uses deductive reasoning to reach a valid conclusion is called a proof. In one type of proof, a paragraph proof, you write a paragraph to explain why a statement is true. A statement that can be proved true is called a theorem. You can use undefined terms, definitions, postulates, and already-proved theorems to prove other statements true. −− Example In △ABC, BD is an angle bisector. Write a paragraph proof to show that ∠ ABD # ∠CBD. By definition, an angle bisector divides an angle into two congruent −−− angles. Since BD is an angle bisector, ∠ ABC is divided into two congruent angles. Thus, ∠ ABD $ ∠CBD. B A D C Exercises 1. Given that ∠ A $ ∠D and ∠D $ ∠E, write a paragraph proof to show that ∠ A $ ∠E. B C M E −−− −− 3. Given that S is the midpoint of QP, T is the midpoint of PR, −− and P is the midpoint of ST, write a paragraph proof to show that QS = TR. Chapter 2 005_054_GEOCRMC02_890511.indd 32 32 N F R Q S P T Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. −−− −− −−− 2. It is given that BC $ EF, M is the midpoint of BC, and N is the −− midpoint of EF. Write a paragraph proof to show that BM = EN. Glencoe Geometry PDF Pass 4/10/08 2:15:24 PM NAME 2-5 DATE PERIOD Skills Practice Postulates and Paragraph Proofs Explain how the figure illustrates that each statement is true. Then state the postulate that can be used to show each statement is true. F C B 1. Planes O and M intersect in line r. N A O E p D M G H 2. Line p lies in plane N. N J I q K L r s Determine whether each statement is always, sometimes, or never true. Explain your reasoning. 3. Three collinear points determine a plane. 4. Two points A and B determine a line. ! # and DP ! #. State # is in plane J and H lies on DG In the figure, DG the postulate that can be used to show each statement is true. 6. G and P are collinear. H P J D G 7. Points D, H, and P are coplanar. −− 8. PROOF In the figure at the right, point B is the midpoint of AC and −−− point C is the midpoint of BD. Write a paragraph proof to prove that AB = CD. A B C D Lesson 2-5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 5. A plane contains at least three lines. Chapter 2 005_054_GEOCRMC02_890511.indd 33 33 Glencoe Geometry PDF Pass 5/13/08 5:55:30 PM NAME 2-5 DATE PERIOD Practice Postulates and Paragraph Proofs Explain how the figure illustrates that each statement is true. Then state the postulate that can be used to show each statement is true. + 3 1. The planes J and K intersect at line m. ℓ 2 2. The lines l and m intersect at point Q. m 1 , Determine whether the following statements are always, sometimes, or never true. Explain. 3. The intersection of two planes contains at least two points. 4. If three planes have a point in common, then they have a whole line in common. " lie in plane A . State the postulate In the figure, line m and TQ that can be used to show that each statement is true. S Q L !"# intersect at T. 6. Line m and ST −− −−− 7. In the figure, E is the midpoint of AB and CD, and AB = CD. Write a −− −−− paragraph proof to prove that AE % ED. C A E B D Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. m A T 5. Points L, and T and line m lie in the same plane. 8. LOGIC Points A, B, and C are noncollinear. Points B, C, and D are noncollinear. Points A, B, C, and D are noncoplanar. Describe two planes that intersect in line BC. Chapter 2 005_054_GEOCRMC02_890511.indd 34 34 Glencoe Geometry PDF Pass CRX 4/9/08 4:52:46 PM