Bellwork

1. Clickers Bellwork • Classify the following angles based on their measure • 102o • 37o • Find the complement and supplement of XYZ if m  XYZ=80 • If XY=YZ, is Y the midpoint of XZ? If not, give a counter example.

2. Bellwork Solution • Classify the following angles based on their measure • 102o • 37o Obtuse Acute

3. Bellwork Solution • Find the complement and supplement of XYZ if mXYZ=80

4. Bellwork Solution • If XY=YZ, is Y the midpoint of XZ? If not, give a counter example.

5. Analyze Conditional Statements Section 2.2

6. The Concept • Today we’re going to revisit a small topic from algebra. • This topic is important to us because it develops a format for us to use our understanding of inductive reasoning

7. Definition • Conditional Statement • Logical statement that uses a hypothesis and a conclusion Axis of symmetry Vertex

8. If-Then statements • Most conditional statements follow an if-then format • The if part of the statement is the hypothesis and the then portion is the conclusion If you took good notes, then the test was easier Hypothesis Conclusion Axis of symmetry Vertex

9. Translating into if-then form The ability to translate a statement into if-then form depends on the ability to see the two statements within one context For instance All whales are mammals. If an animal is a whale, then it is a mammal.

10. Quiz Which of the following is the correct conditional statement based on the following regular statement The weather has to be cold for it to snow. • If it is snowing, then the weather is cold. • If it is cold, then it can snow. • It will snow, if the weather is cold.

11. More Definitions • Negation • The opposite of the original statement • Converse • A rewritten conditional statement in which the hypothesis and conclusion are exchanged • Inverse • A rewritten conditional statement in which both the hypothesis and conclusion are negated • Contrapositive • A rewritten conditional statement in which both parts of the converse are negated Axis of symmetry Vertex

12. Examples • Conditional Statement • p→q • If you are a UA student, then you are wearing a polo shirt • Converse • q→p • If you are wearing a polo shirt, then you are a UA student • Inverse • ~p→~q • If you are not a UA student, then you are not wearing a polo shirt • Contrapositive • ~q→~p • If you are not wearing a polo shirt, then you are not a UA student Axis of symmetry Vertex

13. Examples The following statement is which kind of statement based on the conditional, “If the cafeteria serves pizza, then the majority of students will eat it.” If the majority of student’s don’t eat it, then the cafeteria didn’t serve pizza • Converse • Inverse • Contrapositive Axis of symmetry Vertex

14. Examples The following statement is which kind of statement based on the conditional, “If the cafeteria serves pizza, then the majority of students will eat it.” If the majority of student’s eat it, then the cafeteria served pizza • Converse • Inverse • Contrapositive Axis of symmetry Vertex

15. Examples The following statement is which kind of statement based on the conditional, “If the cafeteria serves pizza, then the majority of students will eat it.” If the cafeteria didn’t serve pizza, then the majority of student’s didn’t eat. • Converse • Inverse • Contrapositive Axis of symmetry Vertex

16. Testing Validity • True • False • Statement: Soccer players are athletes • Conditional Statement • If p, then q • If you are a soccer player, then you are an athlete • Converse • If q, then p • If you are an athlete, then you are a soccer player • Inverse • If not p, then not q • If you are not a soccer player, then you are not an athlete • Contrapositive • If not q, then not p • If you are not an athlete, then you are not a soccer player Axis of symmetry Vertex

17. Equivalent Statements • More often than not, we see a pattern develop with conditional statements • Conditional;True • Inverse; False • Converse; False • Contrapositive;True • In these situations, the conditional and contrapositive are called equivalent statements • The inverse and converse are also • This is important as it relates to writing definitions Axis of symmetry Vertex

18. Equivalent Statements • For example • If two lines intersect to form a right angle, then they are perpendicular lines • The contrapositive is also true • If two lines are not perpendicular, then they do not intersect to form a right angle • Can we make a similar statement to define our understanding of parallel lines and how they intersect? • If two lines do not intersect, then they are parallel • Contrapositive: If two lines are not parallel, then they do intersect Axis of symmetry Vertex

19. Example • What statement can we make about this picture 2 1 m1+ m2=180o Axis of symmetry Vertex

20. Bi-Conditional Statements Does something special happen when both the conditional and its converse are true? Definition: Bi-Conditional Statement: Special conditional statement possible when both the conditional and the converse are true; denote by the phrase, “if and only if”

21. Bi-Conditional Statements T For example: Conditional: If a plastic can be recycled in Kansas City, then it must be a #1 or #2 T Converse: If plastic bottle is a #1 or #2, then it can be recycled in Kansas City Bi-Conditional: A plastic can be recycled in Kansas City, if and only if it is a #1 or #2

22. Examples • What could we say about our previous example? 2 1 m1+ m2=180o Axis of symmetry Vertex

23. On your own • Can the following conditional statement be transformed into a bi-conditional statement. If you can see outside, you will see the sunshine • Yes • No Axis of symmetry Vertex

24. On your own • Can the following conditional statement be transformed into a bi-conditional statement. If you’re not going to eat your vegetables, then you’re not going to grow up to be big and strong • Yes • No Axis of symmetry Vertex

25. Homework • 2.2 • 1-21, 25, 31, 32 • 4th hour: 1-6, 8-20 even, 25, 31

26. Most Important Points • Conditional Statements • Negations • Converse • Inverse • Contrapositive