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## Warm-up

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**Warm-up**• Get all papers passed back • Get out a new sheet for ch. 6 warm-ups • Solve .. • x + 2 > 5 • x – 9 < 7 • 2x > 10 • 10 < -5x Write down one rule that is used when dealing with inequalities. Think back to algebra.**6-1 Inequalities**Objectives • Apply properties of inequality to positive numbers, lengths of segments, and measures of angles • State and use the Exterior Angle Inequality Theorem.**Law of Trichotomy**• The "Law of Trichotomy" says that only one of the following is true**Alex Has Less Money Than Billy or**• Alex Has the same amount of money that Billy has or • Alex Has More Money Than Billy Makes Sense Right !**Equalities vs Inequalities**• To this point we have dealt with congruent • Segments • Angles • Triangles • Polygons**Equalities vs Inequalities**• In this chapter we will work with • segments having unequal lengths • Angles having unequal measures**The symbol "points at" the smaller value**Does a < b mean the same as b > a?**A review of some properties of inequalities (p. 204)**• When you use any of these in a proof, you can write as your reason, A property of Inequality**Example**If a < b, then a + c < b + c Alex has less coins than Billy. • If both Alex and Billy get 3 more coins each, Alex will still have less coins than Billy.**Likewise**• If a < b, then a − c < b − c • If a > b, then a + c > b + c, and • If a > b, then a − c > b − c So adding (or subtracting) the same value to both a and b will not change the inequality**3. If a < b, and c is negative, then ac > bc (inequality**swaps over!)**This is true for division also !**• If a < b, and c is positive, then a < b c cIf a < b, and c is negative, then a > b c c • Who can provide an example of this inequality?**Example**If a < b and b < c, then a < c 1.) If Alex is younger than Billy and 2.) Billy is younger than Carol, Then Alex must be younger than Carol also!**5. If a = b + c and b & c are > 0, then a > b and a > c**Why ?**The Exterior Angle Inequality Theorem**• The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle. • Remember the exterior angle theorem? Based on the diagram, mL4 = _____ + ______ 2 m 4 > m 1 m 4 > m 2 1 3 4**If a and b are real numbers and a < b, which one of the**following must be true? A. -a < -b B. -a > -b C. a < -b • -a > b • I don’t know**True or False**• If XY = YZ + 15, then XY > YZ • If m A = m B + m C, then m B > m C • If m H = m J+ m K, then m K > m H • If 10 = y + 2, then y > 10**White Board Practice**Given: RS < ST; ST< RT Conclusion: RS ___ RT R S T**White Board Practice**Given: RS < ST; ST< RT Conclusion: RS < RT R S T**White Board Practice**Given: m PQU = m PQT + m TQU Conclusion: m PQU ____ m TQU m PQU ____ m PQT U T R Q P**White Board Practice**Given: m PQU = m PQT + m TQU Conclusion: m PQU >m TQU m PQU >m PQT U T R Q P**6-2: Inverses and Contrapositives**• State the contrapositives and inverse of an if-then statement. • Understand the relationship between logically equivalent statements. • Draw correct conclusions from given statements.**Warm – up**• Identify the hypothesis and the conclusion of each statements. Then write the converse of each. • If Maria gets home from the football game late, then she will be grounded. • If Maria is grounded, then she got home from the football game late. • If Mike eats three happy meals, then he will have a major stomach ache. • If Mike has a major stomach ache, then he ate three happy meals.**Venn Diagrams**ALL IF/THEN STATEMENTS CAN BE SHOWN USING A VENN DIAGRAM. THEN • Geographical boundaries are created • Take the statement and put the hypothesis and conclusion with in these boundaries • Example – Think of a fugitive and his whereabouts – City/State IF**She will be grounded**Maria gets home from the game late Venn Diagrams If Maria gets home from the football game late, then she will be grounded.**He will have a major stomach ache**Mike eats three happy meals Venn Diagrams If Mike eats three happy meals, then he will have a major stomach ache.**THEN q.**IF p, Venn Diagrams If we have a true conditional statement, then we know that the hypothesis leads to the conclusion.**Logically Equivalent**Conditional Contrapositve • Inverse • Converse THESE STATEMENTS ARE EITHER BOTH TRUE OR BOTH FALSE!!!**He will have a major stomach ache**Mike eats three happy meals Conditional / ContrapostiveLogically Equivalent If Mike eats three happy meals, then he will have a major stomach ache. If Mike did not have a major stomach ache, then he did not eat three happy meals.**It’s a funny thing**• This part of geometry is called LOGIC, however, if you try and “think logically” you will usually get the question wrong. • Let me show you**Venn Diagrams**What do the other colored circles represent? THEN IF**Aren’t there other reasons why Maria might get grounded?**Then she is grounded Late from football game**Aren’t there other reasons why Mike might get a stomach**ache? Has a major stomach ache Eats three happy meals**Example 1**If it is snowing, then the game is canceled. What can you conclude if I say, the game was cancelled?**Example 1**Nothing ! If it is snowing, then the game is canceled. What can you conclude if I say, the game was cancelled?**There are other reasons that the game would be cancelled**Game cancelled D B Snowing A C**All you can conclude it that it MIGHT be snowing and that**isn’t much of a conclusion.**Let’s try again**• Remember don’t think logically. Think about where to put the star in the venn diagram.**Example 2**If you are in Coach Goss’s class, then you have homework every night. a) What can you conclude if I tell you Jim has homework every night?**Jim might be in Coach Goss’s classNo Conclusion**Homework every night D A Coach Goss’s class B C**Example 3**If you are in coach Goss’s class, then you have homework every night. b) What can you conclude if I tell you Rob is in my 2nd period?**Rob has homework every night**Homework every night D A Coach Goss’s class B C**Example 4**If you are in Coach Goss’s class, then you have homework every night. b) What can you conclude if I tell you Bill has Mr. Brady?