2.1 If Then Statements

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1 Chapter Deductive Reasoning Learn deductive logic Do your first - column proof New Theorems and Postulates **PUT YOUR LAWYER HAT ON!!. If Then Statements Recognize the hypothesis and conclusion of an ifthen statement State the converse of an if-then statement Use a counterexample Understand if and only if The If-Then Statement Conditional:is a two part statement with an actual or implied if-then. hypothesis If p, then q. p ---> q conclusion If I play football, then I am an athlete. Circle the hypothesis and underline the conclusion If a = b, then a + c = b + c All theorems, postulates, and definitions are conditional statements!! Hidden If-Thens A conditional may not contain either if or then! Two intersecting lines are contained in exactly one plane. Which is the hypothesis? two lines intersect Which is the conclusion? exactly one plane contains them The whole thing: If two lines intersect, then exactly one plane contains them. (Theorem ) Definition of Converse A conditional with the hypothesis and conclusion reversed. Original: If I play football, then I am an athlete. If q, then p. q ---> p hypothesis conclusion If I am an athlete, then I play football. **BE AWARE, THE CONVERSE IS NOT ALWAYS TRUE!! Definition of Counterexample Using the same hypothesis as the statement, but coming to a different conclusion. If x > 5, then x = 6. The Counterexample x could be equal to 5.5 or 7 etc If I am in 9 th grade, then I am a freshman. always true, no counterexample **Definitions, Theorems and postulates have no counterexample. Otherwise they would not be true. To be true, it must always be true, with no exceptions. Is the original statement T/F Then write the converse and tell if T/F If I play the tuba, then I am in the band. If I am in the band, then I play tuba. If is part of Segment AB, then its part of Line AB. If is part of Line AB, then its part of Segment AB.

2 Provide a counterexample to show that each statement is false. If a number is divisible by, then it is divisible by 6. Provide a counterexample to show that each statement is false. If x = 8, then x = 9. WARM UP Is the original statement T or F? Then write the converse if false, provide a counter example. If points are in line, then they are colinnear. If points are colinnear, then they are in line. If I live in Los Angeles, then I live in CA. If I live in CA, then I live in Los Angleles. False, you could live in San Diego. Properties from Algebra Properties from Algebra Properties of Equality Numbers, variables, lengths, and angle measures WHAT I DO TO ONE SIDE OF THE EQUATION, I MUST DO Do your first proof Use the properties of algebra and the properties of congruence in proofs see properties on page 7 Read the first paragraph This lesson reviews the algebraic properties of equality that will be used to write proofs and solve problems. We treat the properties of Algebra like postulates Meaning we assume them to be true Addition Subtraction Multiplication Division Substitution Add prop of = Subtr. Prop of = Multp. Prop of = Div. Prop of = Substitution Reflexive x = x. Reflexive Prop. Rules of Thumb. Whiteboards Transitive AB AB if x = y and y = z, then x = z. If AB CD and CD EF, then AB EF Transitive Pop. Measurements are = (prop. of equality) Page 0 # s 0 Figures are (Prop. of congruencey)

3 Given: x + 7-8x = Prove: x = - Your First Proof (specifics) STATEMENTS (general rules) REASONS. x + 7-8x =. Given. -5x + 7 =. Substitution. -5x = 5. Subtraction Prop. =. x = -. Division Prop. = Reasons Used in Proofs (pg. 5) Given Information Definitions (bi-conditional) Postulates Properties of equality and congruence Theorems Given : L = 50 Prove: L = 0 Given : Z = 0 YZ = 7 Prove: Y = Y Z Given : L = L Prove: L + L = 80 Discuss #9 page Given : W = YZ / Prove: W = Y Y is the midpoint of Z W Y Z Warm-up. Statements. Y is the midpoint of Z Reasons. Given. Y = YZ. Def of midpoint. W = YZ. Given Page. 0 # Discuss with class Use the Midpoint Theorem and the Bisector Theorem Know the kinds of reasons that can be used in proofs. W = Y. Substitution

4 Being a lawyer When making your case, you might reference laws, statutes, and/or previous cases in order to make your argument YOU BETTER MAKE SURE YOU ARE REFERECING THE CORRECT ONES OR THE JUDGE WILL KICK YOU OUT OF THE COURTROOM!! The Difference between Midpoint Definition and Midpoint Theorem DEFINITION: If M is the midpoint of AB, then AM = MB. THEREOM: If M is the midpoint of AB, then AM = ½ AB and MB = ½ AB The Angle Bisector Theorem Definition: If B is the bisector of ABC, then. If B is the bisector of ABC, then m AB = ½ m ABC m BC = ½ m ABC B A C Given : W = YZ / Y is the midpoint of Z Pg. 5 # -9 Whiteboards is the midpoint of segment AB. A = 0 AB = y 0 Set up an equation and solve for y LABC = 0 degrees Ray B bisects LABC mlab = 0x 0 Solve for x Prove: W = Y Statements W. Y is the midpoint of Z Reasons. Given. Y = YZ. Def of midpoint. W = YZ. Given Y Z. W = Y. Substitution QUIZ REVIEW WARM UP. Special Pairs of Angles Underline the hypothesis and conclusion in each statement Write a converse of each statement and tell whether it is true or false Provide a counter example to show that the statement is false Be able to complete a proof Name the reasons used in a proof (there are 5) Answer true or false. If false, write a one sentence explanation.. The converse of a true statement is sometimes false.. Only one counterexample is needed to disprove a statement.. Properties of equality cannot be used in geometric proofs.. Postulates are deduced from theorems. 5. Every angle has only one bisector. Apply the definitions of complimentary and supplementary angles State and apply the theorem about vertical angles

5 Complimentary & Supplementary angles Rules that apply to either type... We are always referring to a pair of angles ( angles).. No more no less. Angles DO NOT have to be adjacent. **Do not get confused with the angle addition postulate Definition :Complimentary Angles If two angles add up to 90, then they are complimentary. If m B + m = 90, then B and are complimentary. B is the complement of B 50 is the complement of B 0 Definition: Supplementary Angles If two angles add up to 80, then the angles are supplementary. If m B + m = 80, then B and are supplementary. B 97 B is the supplement of? is the supplement of B Complimentary & Supplementary angles Rules that apply to either type... We are always referring to a pair of angles ( angles).. No more no less. Angles DO NOT have to be adjacent. **Do not get confused with the angle addition postulate. In proofs, you must first prove two L s add up to 90 or 80 before saying they are comp or suppl. NEED TO BE EPLICT!! True or False m A + m B + m C = 80, then A, B, and C are supplementary. A- Sometimes B Always C - Never Two right angles are complementary. Vertical Angles Two angles formed on the opposite sides of the intersection of two lines. **THIS THEOREM WILL BE USED IN YOUR PROOFS OVER AND OVER Theorem Vertical angles are congruent (The definition of Vert. angles does not tell us anything about congruency this theorem proves that they are.) White Board Practice Find the measure of a complement and a supplement of T. m T = 89 The only thing the definition does is identify what vertical angles are NEVER USE THE DEFINITION IN A PROOF!!! 5

6 If and are vertical angles, m = x+8 and m = x+, Find x. White Board Practice A supplement of an angle is three times as large as a complement of the angle. Find the measure of the angle. Let x = the measure of the angle. 80 x : This is the supplement 90 x : This is the complement 80 x = (90 x) 80 x = 70 x x = 90 x = 5 Whiteboard Warm Up Student will complete # from page 5 on front board.5 Perpendicular Lines Recognize perpendicular lines Use the theorems about perpendicular lines Definition: Perpendicular Lines ( ) If two intersecting lines form right angles, then they are perpendicular. If two intersecting lines are perpendicular, then they form right angles l m If l m, then the angles are right. If the angles are right, then l m. What can you conclude about the rest of the angles in the diagram and why? Perpendicular Lines ( ) Two lines that form one right angle form four right angles The definition applies to intersecting rays and segments The definition can be used in two ways (biconditional) PG. 56 Page 57 #, 6, 9, White Boards White Boards Line AB Line CD. E A C G D B F 6

7 Proof practice Discover the steps used to plan a proof Given: m = m Prove: m = m Given: m = m Prove: m + m = 80 Given : L = L Prove: L + L = 80 7