LESSON 2 5 CHAPTER 2 OBJECTIVES
POSTULATE a statement that describes a fundamental relationship between the basic terms of geometry. THEOREM a statement that can be proved true. PROOF a logical argument in which each statement you make is supported by a statement that is accepted as true. PARAGRAPH PROOF or INFORMAL PROOF a type of proof explaining why a conjecture for a given situation is true.
EXAMPLES 1) Jesse is setting up a network for his mother's business. There are five computers in her office. He wants to connect each computer to every other computer so that if one computer fails, the others are still connected. How many connections does Jess need to make? 10 Determine whether each statement is always, sometimes, or never true. 2) If points A, B, and C lie in plane M, then they are collinear. sometimes 3) There is exactly one plane that contains noncollinear points P, Q, and R. always 4) There are at least two lines through points M and N. never 5) There is exactly one plane that contains points A, B, and C. sometimes 6) Points E and F are contained in exactly one line. always 7) Two lines intersect in two distinct points M and N. never
LESSON 2 5 POSTULATES and PARAGRAPH PROOFS Determine the number of segments that can be drawn connecting each pair of points. 1) 2) 3) There are 7 computers, and each computer is connected to 6 other computers. Determine whether each statement is always, sometimes, or never true. 4) A line contains exactly one point. 5) Noncollinear points R, S, and T are contained in exactly one plane. 6) Any two lines m and n intersect. 7) If points A, B, and C are noncollinear, then AB, BC, and CA are contained in exactly one plane. 8) Planes R and S intersect in point T. 9) If points G and H are contained in plane M, then GH is to plane M. In the figure, AC and DE are in plane Q and AC DE. Write the postulate that can be used to show each statement is true. 10) Exactly one plane contains points F, B, and E. 11) BE lies in plane Q. Write the postulate that can be used to show each statement is true. 12) The planes J and K intersect at line m. 13) The lines l and m intersect at point Q. HOMEWORK on LESSON 2 5 p92 12 27 all
LESSON 2 6 CHAPTER 2 OBJECTIVES
TWO COLUMN PROOF or FORMAL PROOF a type of proof containing statements and reasons organized in two columns.
ALGEBRA PROPERTIES equal to itself. a = a PQ = PQ If a = b, then b = a If JK = 12, then 12 = JK If a = b and b = c, then a = c If XY = PQ and PQ = 48, then XY = 48 Substitution Property If a = b and c = b, then a = c If AB = CD and 15 = CD, then AB = 15 Addition Property If a = b, then a + c = b + c If PQ 5 = JK, then PQ 5 + 5 = JK + 5 Subtraction Property If a = b, then a c = b c If x + 10 = 4, x + 10 10 = 4 10 Multiplication Property If a = b, then a(c) = b(c) If 1AB = 1CD, then AB = CD 2 2 Division Property If a = b, then a = b c c If 3(PQ) = 18, then PQ = 6 Distributive Property If a(b + c), then ab + ac If 6(x + 4) = 12, then 6x + 24 = 12
EXAMPLES 1) If x + 2 = 9, then x = 7 2) If LM = 8, then 8 = LM 3) RT = RT 4) Subtraction Property Symmetric Property Reflexive Property 5)
6)
Division Property Multiplication Property Transitive Property Addition Property
LESSON 2 6 ALGEBRAIC PROOF I. Write the property that justifies each statement. 1) 2) 3) 4) 5) 6) RS = RS 7) 8) Symmetric Property Transitive Property Transitive Property Multiplication Property Substitution Property Reflexive Property Addition Property Transitive Property
II. Complete the following proof. 9) STATEMENTS REASONS a) a) Given b) b) Multiplication Property c) 6x 21 = x 6 c) d) 5x 21= 6 Distributive Property d) Subtraction Property Addition Property e) 5x = 15 e) f) x = 3 f) Division Property 10) STATEMENTS REASONS a) a) given b) 2 b) Multiplication Property c) 4x + 6 = 18 c) Substitution Property d) 4x + 6 6 = 18 6 d) Subtraction Property e) 4x = 12 e) Substitution f) f) Division Property g) g) Substitution 11) x = 3 STATEMENTS REASONS Given Subtraction Property 3x + 8 8 = 2 8 3x = 6 x = 2 Subtraction Property Division Property 12) STATEMENTS REASONS HOMEWORK on LESSON 2 6 Given Given Transitive Property p97 p98 14 26 all
Geometry deals with numbers as measures, so geometric proofs use properties of numbers. Here are some of the algebraic properties used in proofs. EXAMPLES: 1) STATEMENTS REASONS a) a) Given b) b) Transitive Property 2) Given: PQ QS and QS ST Prove: PQ = ST P S Q T STATEMENTS REASONS a) PQ QS, QS ST a) Transitive Property b) PQ ST b) c) PQ = ST c) Definition of segments
OBJECTIVES CHAPTER 2 LESSON
EXAMPLES:
LESSON 2 7 PROVING SEGMENT RELATIONSHIPS Justify each statement with a property of equality, a property of congruence, or a postulate. 1) QA = QA 2) If AB BC and BC CE, then AB CE. 3) If Q is between P and R, then PR = PQ + QR 4) If AB + BC = EF + FG and AB + BC = AC, then EF + FG = AC. 5) If DE GH, then GH DE 6) If AB RS and RS WY, then AB WY. 7) RS RS Complete each proof. 8) Given: BC = DE Prove: AB + DE = AC 9) Given: Q is between P and R R is between Q and S, PR = QS Prove: PQ = RS 10) Given: SU LR, TU LN Prove: ST NR HOMEWORK on LESSON 2 7 p104 12 21 all p106 31 38 all