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Chapter 2: 2-1: Deductive Reasoning If - Then Statements IF - THEN STATEMENT: a statement whose basic form is If also known as example: (page 32) (page 33) , then . 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. example: CONCLUSION (q): a statement that follows the “ ” in an “If - then” statement. example: p⇒q Forms for Conditional Statements: 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. example: 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 . example: Conditional: If it is snowing, then it is cold. Converse: If p , then q . , then If . If , then . Circle one: TRUE or FALSE example: Conditional: If angle X is acute, then the measure of angle X equals 60 degrees. Converse: , then If . Circle one: TRUE or FALSE You can not assume a converse is just because the original statement is true. COUNTEREXAMPLE: an example used to prove that an if-then statement is The hypothesis is and the conclusion is NOTE: only example: . . 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 example: 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: 1 to 27 odd #’s . 2-2: Properties from Algebra 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: (page 37) 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 ! g. If ∠A ! ∠B, then ∠B ! ∠A. CD and CD ! EF , then AB ! EF . h. If m∠K = 90º, then ∠K is a right angle. Solve for “x” and supply reasons for each step: Steps 1. 5(3x-4)=5x Reasons 1. 2. 2. 3. 3. 4. 4. 5. 5. Complete each of the following proofs. Given: m∠1 = m∠3 D E Prove: m∠ABE = m∠DBC 1 A Statements 1. 2. 2. 3. 3. + m∠DBC = + B 3 C Reasons 1. 4. m∠ABE = 2 Given 4. 5. 5. Given: KP = ST ; PR = TV K P R S T Prove: KR = SV Statements Reasons 1. 1. 2. 2. 3. KP + PR = 3. Given ST + TV = 4. 4. Assignment: Written Exercises, pages 41 & 42: 1 to 9 odd numbers V Proving Theorems 2-3: (page 43) What is a theorem? THEOREM 2-1 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: Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8. example: Given R is the midpoint of SQ. Give the reason that justifies each statement. ! RQ (a) SR (b) SR = 1/2 SQ S R (c) SR + RQ = SQ (d) PR bisects SQ _ P Q THEOREM 2-2 ANGLE BISECTOR THEOREM !!!" If BX is the bisector of ∠ABC, then: m∠ABX = and m∠ABC m∠XBC = m∠ABC. !!!" Given: BX is the bisector of ∠ABC. A Prove: X B C Proof: Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. example: !!!" Given: FD bisects ∠CFE. Give the reason that justifies each statement. C (a) m∠CFD = ½ m∠CFE (b) m∠CFD = m∠DFE (c) CD + DE = CE D F Reasons Used in Proofs: (1) (2) (3) (4) Given information Definitions Postulates (including properties from algebra) Theorems - only ones that have already been proved! E DEDUCTIVE REASONING: proving statements by reasoning from accepted postulates, , , and given information. example: NOTE: Definitions can be written as biconditionals (combine conditional and converse),ie. the conditional and converse are both . Conditional: Converse: If an angle is a right angle,then its measure is 90º. 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. example: Given: M is the midpoint of PQ N is the midpoint of RS PQ = RS P M Q Prove: PM = RN R N S Proof: Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. Assignment: Written Exercises, page 46: 1-2 all Worksheet on Lesson 2-3 Prepare for Quiz on Lessons 2-1 to 2-3: Using Deductive Reasoning 2-4: Special Pairs of Angles (page 50) COMPLEMENTARY ANGLES: two angles whose measures have the sum degrees. example: SUPPLEMENTARY ANGLES: two angles whose measures have the sum degrees. example: examples: B A (1) X C In the diagram, ∠CXD is a right angle. E Name another right angle: 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 + 15. 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 + 20. Find the value of x, m∠P, and m∠K. x = m∠P = m∠K = (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. = = VERTICAL ANGLES: two angles whose sides form two pairs of rays. example: 1 4 2 3 THEOREM 2-3 Vertical angles are congruent. Given: ∠1 and ∠2 are vertical angles. 3 1 Prove: Proof: Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 2 examples: Find the value of “x”. (4) (5) 35º 35º (2x-5)º xº x= x= (6) (7) (x+30)º (2x+30)º (x+y)º (4x)º (4y)º x= 120º x= y= (8) Find the measure of each angle. C D 60º A B G Assignment: Written Exercises, pages 52 & 53: 1-18 all #’s, 19, 21, 25 to 31 odd #’s Worksheet on Lesson 2-4 Prepare for Quiz on Lesson 2-4: Special Pairs of Angles E F Perpendicular Lines 2-5: (page 56) PERPENDICULAR LINES (⊥−lines): two lines that intersect to form angles. example: 1 2 4 3 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 2-4 If two lines are perpendicular, then they form adjacent angles. l Given: l ⊥ n Prove: , , are congruent angles. 2 1 3 4 ,& Proof: Statements Reasons 1. 1. 2. 2. 3. 3. n THEOREM 2-5 If two lines form congruent adjacent angles, then they are . l Given: ∠1 ≅ ∠2 Prove: l ⊥ n 2 1 n Proof: Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. THEOREM 2-6 If the exterior sides of two adjacent acute angles are , then the angles are complementary. !!!" !!!" OA ! OC Given: A Prove: _________________________ B Proof: O Statements C Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. example: Name the definition or theorem that justifies each statement about the diagram. B 1 2 3 4 5 6 7 8 A C 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. !!!" !!!" example: If ZW ! ZY , m∠1 = 5 x, and m∠2 = 2 x - 1, find the value of “x”. x = ________________ W V 1 2 Z Y Assignment: Written Exercises, pages 58 & 59: 3-12 all #’s, 19-25 odd #’s 2-6: 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 the that lead from 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 ? Suggestions for Writing Proofs: (1) Follow one of the suggested methods. (2) (3) (4) Study Use a pencil. TRY, TRY, TRY!!! proofs. ? ”. ”. And so on, until you have a plan. THEOREM 2-7 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; ∠2 ≅ ∠4 1 2 Prove: 3 4 Proof: Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. If two angles are supplements of congruent angles then the two angles are congruent. If two angles are supplements of the same angle then the two angles are congruent. example: Justify each statement. Given: ∠2 is supplementary to ∠3 1 2 3 4 m∠1 + m ∠2 = 180º _____________________________ ∠1 & ∠2 are supplementary angles _____________________________ ∠1 ≅ ∠3 _____________________________ THEOREM 2-8 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; ∠2 ≅ ∠4 1 2 3 4 Prove: Proof: Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. If two angles are complements of congruent angles then the two angles are congruent. If two angles are complements of the same angle then the two angles are congruent. example: Given: LM ! MN and KN ! MN M N 1 a. Name a complement of ∠2: 2 3 4 b. Name a complement of ∠3: L c. What theorem justifies the answers to a & b? d. If ∠2 ≅ ∠3, what can be concluded and why? Assignment: Written Exercises, page 63: 1-14 all #’s Prepare for Quiz on Lessons 2-4 to 2-6 Prepare for Test on Chapter 2: Deductive Reasoning K