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Lesson 2.1 Conditional Statements You will learn to… * recognize and analyze a conditional statement * write postulates about points, lines, and planes using conditional statements A conditional statement has two parts, a hypothesis and a conclusion. pq If p, then q. hypothesis (p) If the team wins the game, then they will win the tournament. conclusion (q) Write an if-then statement. 1. The intersection of two planes is a line. If two planes intersect, then their intersection is a line. Write an if-then statement. 2. A line containing two given points lies in a plane if the two points lie in the plane. If two points lie in a plane, then the line containing them lies in the plane. The negation of a statement is formed by negating the statement. The negation is written ~ p. Write the negation of this statement. 4. m A = 125° m A 125° 5. A is not obtuse A is obtuse The inverse is formed by negating the hypothesis and the conclusion. The inverse is ~ p ~ q. If ~ p, then ~ q. Write the inverse of this if-then statement. Is it true or false? 6. If m A = 125°, then A is obtuse. If m A 125°, then A is not obtuse. False The converse is formed by switching the hypothesis and conclusion. The converse is q p. If q, then p. Write the converse of this if-then statement. Is it true or false? 3. If m A = 125°, then A is obtuse. If A is obtuse, then m A = 125°. False The contrapositive is formed by negating the hypothesis and conclusion of the converse. The contrapositive is ~ q ~ p. If ~ q, then ~ p. Write the contrapositive of this ifthen statement. Is it true or false? 7. If m A = 125°, then A is obtuse. If A is not obtuse, then m A 125°. True Postulate 5 Through any two points there exists exactly one line. Postulate 6 A line contains at least two points. Postulate 7 If two lines intersect, then their intersection is exactly one point. Postulate 8 Through any three noncollinear points there exists exactly one plane. B A T C Postulate 9 A plane contains at least three noncollinear points. Postulate 10 If two points lie in a plane, then the line containing them lies in the plane. Postulate 11 If 2 planes intersect, then their a line intersection is ___________. Lesson 2.2 Biconditional Statements You will learn to… * recognize and use definitions * recognize and use biconditional statements All definitions can be interpreted “forward” and “backward.” All definitions are biconditional. For example, perpendicular lines are defined as two lines that intersect to form one right angle. If two lines are perpendicular, then they intersect to form one right angle. If two lines intersect to form one right angle, then they are perpendicular. A biconditional statement contains the phrase “if and only if.” Two lines are perpendicular if and only if they intersect to form one right angle. A biconditional statement is true when the original if-then statement AND its converse are both true. 1. Two angles are supplementary if and only if the sum of their measures is 180°. if-then statement: If two angles are supplementary, then the sum of their measures is 180°. converse: If the sum of the measures of two angles is 180°, then they are supplementary. 2. If an angle is 135˚, then it is an obtuse angle. converse: If an angle is obtuse, then its measure is 135°. Can we write a biconditional statement? counterexample? 3. If two angle measures add up to 90˚, then they are complementary angles. converse: If two angles are complementary, then the sum of their measures is 90°. Can we write a biconditional statement? Two angles are complementary if and only if the sum of their measures is 90°. Workbook Page 25 (1-7) Lesson 2.3 Deductive Reasoning You will learn to… * use symbolic notation to represent logical statements * form conclusions by applying laws of logic Using these phrases, write the conditional statement. p: mB = 90˚ q: B is a right angle 1. p q 2. q p 3. ~ p ~ q 4. ~ q ~ p 5. p q If mB = 90˚, If B is a right then B is a Ifangle, mB then ≠ 90˚, right angle. If mB Bis= a then Bnot is not 90˚ =angle. 90˚ if right angle, then amB right and only if B mB ≠ 90˚ is a right angle. Deductive Reasoning uses facts to make a logical argument. definitions, properties, postulates, theorems, and laws of logic Law of Detachment Given facts Therefore: pq p q hypothesis is true conclusion must be true You can use these symbols when asked to explain your reasoning. Law of Detachment q p If I learn my facts, then I will pass geometry. p I learned my facts. q Therefore, I passed geometry. Law of Syllogism p q Given qr facts Therefore: p r You can use these symbols when asked to explain your reasoning. Law of Syllogism p q If I pass geometry, then my dad will be happy. q If my dad is happy, then I will r get a cell phone. p Therefore, if I pass geometry, then I will get a cell phone. r 6. Is this argument valid? If 2 lines in a plane are parallel, then they do pq not intersect. Coplanar lines n and m are parallel. p Therefore, lines n and q m do not intersect. VALID – Law of Detachment 7. Is this argument valid? If 2 angles are supplementary, then the sum of their measures is 180˚. If 2 angles form a linear pair, then they are supplementary. pq rp pr qp Therefore, if 2 angles form r q a linear pair, then the sum of their measures is 180˚ VALID – Law of Syllogism 8. Is this argument valid? If 2 angles are a linear pair, then the sum of their measures is 180˚. m1 + m2 = 180˚ Therefore, 1 and 2 are a linear pair. INVALID pq q p 9. Is this argument valid? If you live in Canada, then you live in North America. rp qq If you live in South Carolina, then you live in r qq p North America. Therefore, if you live in pr Canada, then you live in South Carolina INVALID If you use this product, then you will have even-toned skin. If you have even-toned skin, If you use this product, youyou will will be beautiful. then be beautiful. then Lesson 2.4 Properties of Equality and Congruence You will learn to… * use properties from algebra * use properties of length and measure to justify segment and angle relationships Equality Properties Reflexive Property Symmetric Property Transitive Property Addition Property Subtraction Property Multiplication Property Division Property Distributive Property Substitution Property Reflexive Property Mrs. R is the same height as Mrs. R. A = A Symmetric Property IF Mrs. R is the same height as Mr. S THEN Mr. S is the same height as Mrs. R If A = B , then B = A Transitive Property IF Mrs. R is the same height as Mr. S AND THEN Mr. S is the same height as Mrs. T Mrs. R is the same height as Mrs. T If A = B and B = C, then A = C Reflexive Property XY XY m X m X Symmetric Property If MN 20, then 20 MN If mN mM then mM mN Transitive Property If XY = ST and ST = 10, then XY = 10 If mA = mB and mB = 10°, then mA = 10° Division Property If 8x=16, then x=2. Addition Property If x-7=5, then x=12. Multiplication Property If ½ x = 7, then x=14. Subtraction Property If x+3=7, then x=4. Substitution Property If 2 A=x and x=6, then A=36. If 4 + 7x – 10 = 24, Then 7x - 6 = 24 Distributive Property If B=2(4x + 3), then B=8x + 6. If 4x + 7x = 24, Then 11x = 24 Proofs !! Memorize definitions, postulates, and theorems as we learn them. Write out entire proof each time one is in the assignment. Don’t give up!!!! You can do it!!!! Let’s Practice… 2. 4+2(3x+5)=11-x Given 4+6x+10=11-x Distributive prop. 14+6x =11-x 14 + 7x = 11 Substitution Addition prop. 7x = - 3 x = - 3/7 Subtraction prop. Division Prop. 4. 1/ 5 x + 4 = 2x + 3/5 Given 1x + 20 = 10x + 3 Multiplication Prop 20 = 9x + 3 Subtraction Prop Subtraction Prop 17 = 9x 17/ 9 =X Division Prop 5. Given that MN=PQ, show that MP=NQ MN = PQ MP = MN + NP MP = PQ + NP Q P N Given Segment Addition Postulate NQ = PQ + NP Substitution Prop Segment Addition Postulate MP = NQ Substitution Prop M 7. Given mAQB=mCQD,show that mAQC=mBQD Q mAQB = mCQD Given D A B C mAQB + mBQC = mAQC Angle Addition Postulate mCQD + mBQC = mAQC Substitution Angle Addition mCQD + mBQC = mBQD Postulate mAQC = mBQD Substitution 8. Given mRPS=mTPV and mTPV=mSPT P show that mRPV=3(mRPS) mRPS = mTPV mTPV = mSPT Given R S T V Given mRPS = mSPT Transitive Prop Angle Addition mRPV= mRPS+mSPT+mTPV Postulate mRPV= Substitution mRPS+mRPS+mRPS mRPV = 3(mRPS) Distributive Prop You can use definitions as reasons in proofs. Statements Reasons 1) A is a right angle 1) Given 2) m A = 90˚ 2) Def. of right angles Statements Reasons 1) m A = 90˚ 1) Given 2) A is a right angle 2) Def. of right angles C 1 Statements A Reasons D B 1) AB CD 1) Given 2) 1 is a right angle 2) Def. of lines C 1 A D B Statements 1) 1 is a right angle Reasons 1) Given 2) AB CD 2) Def. of lines 1 2 Statements Reasons 1) 1 and 2 1) Def. of vertical angles are vertical angles 2) 1 2 2) Vertical Angles Theorem 2 Statements 1) 1 and 2 are a linear pair 2) 1 & 2 are Reasons 1) Def. of linear pair 2) Linear Pair supplementary 3) m1 + m2 = 180° 1 3) Postulate Def. of supplementary Statements 1) AB = CD 2) AB CD Reasons 1) Given 2) Def. of segment Statements 1) AB CD Reasons 1) Given 2) AB = CD 2) Def. of segment Statements Reasons 1) m1 = m2 1) Given 2) 1 2 2) Def. of angles Statements 1) 1 2 Reasons 1) Given 2) m1 = m2 2) Def. of Lesson 2.5 Proving Statements about Segments You will learn to… * justify statements about congruent segments * write reasons for steps in a proof use practice sheet of proofs Proofs !! Memorize definitions, postulates, and theorems as we learn them. Write out entire proof each time one is in the assignment. Don’t give up!!!! You can do it!!!! Reflexive Property of Congruence XY XY Symmetric Property of Congruence If XY JK, then JK XY Transitive Property of Congruence If XY JK and JK MN , then XY MN 1. Given: EF = GH Prove: EG FH H G E F (Proof is on next slide) 1. Statements 1) EF = GH 2) EF + FG = GH + FG 3) EG = EF + FG FH = GH + FG 4) EG = FH 5) EG FH Reasons 1) Given 2) Addition Prop. Segment Addition 3) Postulate 4) Substitution 5) Def. of 2. Given: RT WY, ST = WX Prove: RS XY T S R X W Y 2. Statements Reasons 1) Given 1) RT WY 2) RT = WY 2) Def. of 3) RT = RS + ST 3)Segment Addition WY = WX + XY Postulate 4) RS + ST = WX + XY 4) Substitution 5) ST = WX 5) Given 6) RS + ST = ST + XY 6) Substitution 7) RS = XY 7) Subtraction Prop. 8) Def. of 8) RS XY 3. Given: X is the midpoint of MN and MX = RX Prove: XN = RX M S X R N 3. Statements 1) X is the midpoint of MN 2) NX = MX 3) MX = RX 4) NX = RX Reasons 1) Given 2) Def. of midpoint 3) Given 4) Transitive Prop. Paragraph proof example for #1 Since EF = GH, EF + FG = GH + FG by the Addition Property. EG = EF + FG and FH = GH + FG by the Segment Addition Postulate. By Substitution, EG = FH. Therefore, EG FH by the definition of congruent segments. Paragraph proof example for #3 So, I was chillin’ with the homeboys and my homeboy Sherrod tells me, “Dave, x is the midpoint of MN, so NX = MX.” I said, “Sherrod, how do you figure?” Sherrod tells me “The definition of midpoint says so!” So I was like, “yo, Sherrod, did you know MX = RX,” and he said, “really, well then NX = RX Dawg. “Sherrod, my homie, I didn’t know you were so smart,” I said, “how did you figure that out?” He was like, “Substitution, my brother!” David Mathews # 17 Statements 1) XY = 8, XZ = 8, 2) XY = XZ 3) XY XZ 4) XY ZY 5) XZ ZY # 18 Statements 1) NK NL, NK = 13 2) NK = NL 3) NL = 13 Lesson 2.6 Proving Statements about Angles You will learn to… * use angle congruence * prove properties about special pairs of angles Theorem 2.3 Right Angle Congruence Theorem right angles are All ________ congruent __________. A is supplementary to 40° B is supplementary to 40° What do you know about A and B? A B Theorem 2.4 Congruent Supplements Theorem If 2 angles are supplementary to the same angle, then they are congruent _______________. Using the Congruent Supplements Theorem… Statements 1) 1 & 2 are supp. 1 & 3 are supp. Reasons 2) 2 3 2) Congruent Supplements Theorem A is complementary to 50° B is complementary to 50° What do you know about A and B? A B Theorem 2.5 Congruent Complements Theorem If 2 angles are complementary to the same angle, then they are congruent _______________. Using the Congruent Complements Theorem… Reasons Statements 1) 1 & 2 are comp. 1 & 3 are comp. 2) 2 3 2) Congruent Compliments Theorem Postulate 12 Linear Pair Postulate If two angles form a linear pair, then they are supplementary _______________. Using the Linear Pair Postulate… Statements 1) 1 & 2 are a linear pair Reasons 1) Def. of linear pair 2) 1 & 2 are supplementary 2) Linear Pair Postulate 3) m1 + m2 = 180 3) Def. of supplementary Theorem 2.6 Vertical Angles Theorem Vertical angles are congruent _______________. Using the Vertical Angles Theorem… Statements 1) 1 & 2 are vertical angles Reasons 1) Def. of vertical angles 2) 1 2 2) Vertical Angles Theorem 1. Given: 1 2 , 3 4 , 2 3 Prove: 1 4 1 2 4 3 1. Statements 1. 1 2 , 2 3 2. 1 3 3. 3 4 4. 1 4 Reasons 1. Given 2. Transitive Prop. 3. Given 4. Transitive Prop. 2. Given: m1 = 63˚,1 3 , 3 4 Prove: m4 = 63˚ 1 2 3 4 2. Statements 1. m1 = 63˚, 1 3 , 3 4 2. 1 4 3. m1 = m4 4. m4 = 63˚ Reasons 1. Given 2. Transitive Prop. 3. Def of 4. Substitution 3. Given: DAB & ABC are right angles , ABC BCD Prove: DAB BCD D C A B 3. Statements 1. DAB & ABC are right angles 2. DAB ABC 3. ABC BCD 4. DAB BCD Reasons 1. Given 2. All right s are 3. Given 4. Transitive Prop. 4. Given: m1 = 24˚,m3 = 24˚ 1 & 2 are complementary 3 & 4 are complementary Prove: 2 4 1 2 4 3 4. Statements m1 = 24˚, m3 = 24˚ 1. 1 & 2 are comp. 3 & 4 are comp. 2. m1 = m3 3. 1 3 4. 2 4 Reasons 1. Given 2. Substitution 3. Def of 4. Congruent Complements Theorem 5. Statements Reasons 1. 1 and 2 are a linear pair 1. Given 2 and 3 are a linear pair 2. 1 and 2 are supp. Linear Pair 2. 2 and 3 are supp. Postulate 3. 1 3 3. Congruent Supplements Theorem 6. Statements 1. QVW and RWV Reasons 1. Given are supplementary Def. of Linear Pair a linear pair 3. QVW and QVP are 3. Linear Pair Postulate supplementary 4. QVP RWV 4. Congruent 2. QVW and QVP are 2. Supplements Theorem #24 & #26 for homework #24 Statements 1) 3 and 2 are complementary 2) m1 + m2 = 90 3) m 3 + m2 = 90 4) m1 + m2 = m3 + m2 5) m1 = m3 6) 1 3 #26 Statements 1) 4 and 5 are vertical angles 2) 6 and 7 are vertical angles 3) 4 5 , 6 7 4) 5 6 5) 4 7