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
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!
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?