Chapter 2: Deductive Reasoning (page 32) p q


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1 Chapter 2: Deductive Reasoning (page 32) 21: If  Then (page 33) IF  THEN STATEMENT: a statement whose basic form is If, then. also known as statements or. If it is snowing, then it is cold. Conditional statements are either or. HYPOTHESIS (p): a statement that follows the in an If  then statement. CONCLUSION (q): a statement that follows the in an If  then statement. Forms for Conditional : p q If it is snowing, then it is cold. If p, then q. It is snowing implies it is cold. p implies q. It is snowing only if it is cold. p only if q. It is cold if it is snowing. q if p. Circle one: TRUE or FALSE NOTE: If, then, implies, & only if are part of the hypothesis or the conclusion.
2 If angle X is acute, then the measure of angle X equals 60 degrees. implies only if if Circle one: TRUE or FALSE CONVERSE (of a conditional statement): is formed by interchanging the and the conclusion. Converses to conditional statements are either or. Conditional: If it is snowing, then it is cold. If p, then q. Converse: If, then. If, then. Circle one: TRUE or FALSE Conditional: If angle X is acute, then the measure of angle X equals 60 degrees. Converse: If, then. Circle one: TRUE or FALSE You can not assume a converse is just because the original statement is true.
3 COUNTEREXAMPLE: an example used to prove that an ifthen statement is. The hypothesis is and the conclusion is. NOTE: only counterexample is needed to prove a statement false! Two angles are adjacent if they have a common vertex. Conditional: If, then. TRUE or FALSE  Converse: If, then. TRUE or FALSE  BICONDITIONAL: a statement that combines a with the words if and only if. and its Conditional: If two angles are congruent, then their measures are equal. p q Biconditional: if and only if. p q Assignment: Written Exercises, page 35: 3,5,7,9,12,15,16,18,21,24,27,30,31
4 22: Properties from Algebra (page 37) Properties of Equality (=) Addition Property: Subtraction Property: Multiplication Property: Division Property: Substitution Property: Reflexive Property: Symmetric Property: Transitive Property: Distributive Property: Properties of Congruence (!) Reflexive Property: Symmetric Property: Transitive Property:
5 Statement Justification a. If AB = CD and BC = BC, then AB + BC = CD + BC. b. If 2 + YZ = 8, then YZ = 6. c. If 2 m 1 = 72º, then m 1 = 36º. d. If Pt.B is between A and C, then AB + BC = AC. e. If m A = m X and m X = m B, then m A = m B. f. If AB! CD and CD! EF, then AB! EF. g. If A! B, then B! A. h. If m K = 90º, then K is a right angle. Solve for x and supply reasons for each step: Steps 1. 5 ( 3 x  4 ) = 5 x 1.
6 Complete each of the following proofs. Given: m 1 = m 3 D E Prove: m ABE = m DBC A B C Given 4. m ABE = + 4. m DBC = + Given: KP = ST ; PR = TV Prove: KR = SV K P R S T V Given 3. KP + PR = 3. ST + TV = Assignment: Written Exercises, pages 41 & 42: 1 to 13 odd numbers
7 23: Proving Theorems (page 43) What is a theorem? THEOREM 21 MIDPOINT THEOREM If M is the midpoint of AB, then AM = AB and MB = AB. Given: M is the midpoint of AB Prove: A M B Proof: Given R is the midpoint of SQ. Give the reason that justifies each statement. (a) SR! RQ (b) SR = 1 /2 SQ (c) SR + RQ = SQ _ S R (d) PR bisects SQ P Q
8 THEOREM 22 ANGLE BISECTOR THEOREM!!!" If BX is the bisector of ABC, then: m ABX = m ABC!!!" Given: BX is the bisector of ABC. Prove: and m XBC = m ABC. A X Proof: B C Given: FD!!!" bisects CFE. Give the reason that justifies each statement. C (a) m CFD = ½ m CFE D (b) (c) m CFD = m DFE CD + DE = CE F E Used in Proofs: (1) Given information (2) Definitions (3) Postulates (including properties from algebra) (4) Theorems  only ones that have already been proved!
9 DEDUCTIVE REASONING: proving statements by reasoning from accepted postulates,,, and given information. NOTE: Definitions can be written as biconditionals (combine conditional and converse),ie. the conditional and converse are both. Conditional: If an angle is a right angle,then its measure is 90º. Converse: If the measure of an angle is 90º, then the angle is a right angle. Biconditional: An angle is a right angle its measure is 90º. Given: M is the midpoint of PQ N is the midpoint of RS P M Q PQ = RS Prove: PM = RN R N S Proof: Assignment: Written Exercises, pages 46 & 47: 114 all, 19 & 20 Worksheet on Lesson 23 Prepare for Quiz on Lessons 21 to 23: Using Deductive Reasoning
10 24: Special Pairs of Angles (page 50) COMPLEMENTARY ANGLES: two angles whose measures have the sum degrees. SUPPLEMENTARY ANGLES: two angles whose measures have the sum degrees. examples: (1) In the diagram, CXD is a right angle. Name another right angle: B A X C E D Name complementary angles: Name 2 congruent supplementary angles: Name 2 noncongruent supplementary angles: (2) C and D are complementary, m C = 3 x  5 and m D = x Find the value of x, m C, and m D. x = m C = m D = (3) P and K are suppplementary, m P = 4 x + 10 and m K = x Find the value of x, m P, and m K. x = m P = m K =
11 (4) A supplement of an angle is seven times a complement of the angle. Find the measures of the angle, its complement, and its supplement. measure of angle = = measure of comp. = = measure of supp. = = (5) A complement of an angle is 15 less than twice the measure of the angle. Find the measures of the angle, its complement, and its supplement. measure of angle = = measure of comp. = = measure of supp. = =
12 VERTICAL ANGLES: two angles whose sides form two pairs of rays THEOREM 23 Vertical angles are congruent. Given: 1 and 2 are vertical angles Prove: Proof: 6. 6.
13 examples: Find the value of x. (4) (5) 35º (2x5)º xº 35º x = x = (6) (7) (4x)º (x+30)º (2x+30)º (x+y)º (4y)º 120º x = x = y = (8) Find the measure of each angle. C D 60º A B E G F Assignment: Written Exercises, pages 52 to 54: 131 odd # s, Worksheet on Lesson 24 Prepare for Quiz on Lesson 24: Special Pairs of Angles
14 25: Perpendicular Lines (page 56) PERPENDICULAR LINES ( lines): two lines that intersect to form angles m n If m n, then,,, and are right angles. How many angles must be right angles in order for the lines to be perpendicular? THEOREM 24 If two lines are perpendicular, then they form adjacent angles. Given: l n l 2 1 Prove:,,, & 3 4 n are congruent angles. Proof:
15 THEOREM 25 If two lines form congruent adjacent angles, then they are. Given: 1 2 l Prove: l n 2 1 n Proof: THEOREM 26 If the exterior sides of two adjacent acute angles are!!!"!!!" Given: OA! OC A, then the angles are complementary. Prove: B Proof: O C
16 Name the definition or theorem that B justifies each statement about the diagram. 1 2 A C 7 8 D (a) If AB! BC, then ABC is a right angle. (b) If BD! AC, then 3 4. (c) If CD! AD, then 7 and 8 are complementary. (d) If 7 & 8 are complementary, then m 7 + m 8 = 90º. (e) If 4 6, then AC! BD. (f) 4 5. (g) If ABC is a right angle, then m ABC = 90º. If ZW!!!"! ZY!!! ", m 1 = 5 x, and m 2 = 2 x  1, find the value of x. W V x = 1 2 Z Y Assignment: Written Exercises, pages 58 & 59: 311 odd # s, odd # s Prepare for Quiz on Lessons 22 to 25
17 26: Planning a Proof (page 60) A proof of a theorem consists of five (5) parts: 1. of the theorem. 2. A that illustrates the given information. 3. A list, in terms of the figure, of what is. 4. A list, in terms of the figure, of what you are to. 5. A series of and that lead from the information to the statement that is to be. Methods for Writing Proofs: A. as much information as you can. Sometimes what you can see will show you a plan. Reread the given. What does it tell you? Look at the diagram. What other information can you conclude? B. Work. Go to the conclusion, the part you would like to prove. Think: This conclusion would be true if?. And? would be true if?. And so on, until you have a plan. Suggestions for Writing Proofs: (1) Follow one of the suggested methods. (2) Study proofs. (3) Use a pencil. (4) TRY, TRY, TRY!!! NOTES There is usually more than one way to write a proof of a statement. Be sure that your steps flow logically from one step to the next. Some steps are more important than others. Your proof should have enough steps to allow the reader to follow your argument and to see that the statement is indeed true.
18 THEOREM 27 If two angles are of congruent angles (or of the same angle), then the two angles are congruent. Given: 1 and 2 are supplementary; 3 and 4 are supplementary; Prove: Proof: 3 4 Given: 2 is supplementary to 3 Prove: Proof:
19 THEOREM 28 If two angles are of congruent angles (or of the same angle), then the two angles are congruent. Given: 1 and 2 are complementary; 3 and 4 are complementary; Prove: Proof: 3 4 Given: LM! MN and KN! MN M N a. Name a complement of 2: b. Name a complement of 3: c. What theorem justifies the answers to a & b? L K d. If 2 3, what can be concluded and why? Assignment: Written Exercises, pages 63 & 64: 121 odd # s Prepare for Quiz on Lessons 24 to 26 Prepare for Test on Chapter 2: Deductive Reasoning
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