 Unit 2 – Angles

 Thursday, February 12, 2015

 Friday, February 13, 2015

Bellwork – Complete in the first 5 minutes of class

 Tuesday, February 17, 2015 Bellwork Log into kahoot.it -If you do not have an ipad, team up with a friend. Make sure your nickname includes your number for classwork credit! TOP 4 PLAYER EARN 1 HW PASS EACH

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Wednesday, February 18, 2015  INDUCTIVE REASONING

 Find m<z  Tell me WHY you could figure out what the m<z was (What is the RELATIONSHIP?) Bell Ringer 27 o z

 Inductive Reasoning and Conjecturing

 Conjecture: An educated guess  Inductive Reasoning: Looking at several specific situations to arrive at a conjecture Definitions

 If you place the ball in one corner of a 10- foot by 5-foot carom billiard table and shoot it at a 45 o angle with respect to the sides of the table, where will it go, assuming there are no other balls on the table? Use graph paper to draw the path of the ball. Make a conjecture about the path of the ball if it is shot at a 45 o angle from any corner of the table. Example 1

 The ball will strike the long side of the table at its midpoint.  The ball will then bounce off the rail at the same angle.  The ball will continue on a path and touch the opposite corner. Some conjectures are:

I have been to Columbia three times in my life. Every time I go, it is always raining. I think it must always rain in Columbia. Discuss my reasoning here. Discussion

 When a door is open, the angle the door makes with the door frame is complementary to the angle the door makes with the wall. Write a conjecture about the relationship of the measures of the two angles.

 For points A, B, and C, AB = 10, BC = 8, and AC = 5. Make a conjecture and draw a figure to illustrate your conjecture.  Given: Points A, B, and C, AB = 10, BC = 8, and AC = 5  Conjecture: ? Example 2

 For points A, B, C, and D, AB=5, BC=10, CD=8, and AD=12. Make a conjecture and draw a figure to illustrate your conjecture.  Given: ?  Conjecture: ?

 Diamond was driving her friends to school when her car suddenly stopped two blocks away from school. Make a list of conjectures that Diamond can make and investigate as to why her car stopped. Example 3

 Destynee was preparing toast for breakfast. After a few minutes the bread popped up but was not toasted. Make a list of conjectures that Destynee can make as to why the bread was not toasted.

 Counterexample: A false example Definition

 Given that points P, Q, and R are collinear, Joel made a conjecture that Q is between P and R. Determine if his conjecture is true or false.  Given: Points P, Q, and R are collinear.  Conjecture: Q is between P and R. Example 4

Exit Ticket  Explain the meaning of conjecture in your own words.  Describe why three points on a circle could never be collinear.  Explain how you can prove that a conjecture is false.  Determine if the conjecture if true or false based on the given information. Explain your answer.  Given: <1 and <2 are supplementary angles <1 and <3 are supplementary angles.  Conjecture: <2 = <3

Thursday, February 19, 2015

 SWBAT define conditional, the parts of a conditional, converses and counterexamples. TODAY’S OBJECTIVE: Geometry - Lesson 2.2

Conditionals  Conditionals are if-then statements  If-then statements are used to CLARIFY statements that may seem confusing.  Ex: If you like Jay-Z, then you like rap.  Ex: If x=3, then 2x=6.

Parts of Conditionals  Every conditional must have the following:  A hypothesis – the part that is assumed (comes after the word ‘if’)  A conclusion – the part that is concluded (comes after the word ‘then’)

Examples  If today is Monday, then tomorrow is Tuesday.  Hypothesis: today is Monday  Conclusion: tomorrow is Tuesday  If y-3=5, then y=8.  Hypothesis: y-3=5  Conclusion: y=8

Symbolic Form  To write a conditional quickly, we use symbols and letters to represent parts of the conditional.  The hypothesis is represented by the letter ‘p’  The conclusion is ‘q’  The ‘if-then’ is   Example p  q is read “if p, then q”

Is every conditional true?  NO! We can determine the ‘truth value’ (whether the statement is true or false) for every conditional.  If you can find a counterexample for the conditional, then the conditional is false (only 1 counterexample is needed to prove a conditional false).

Counterexamples  If you live in Pennsylvania, then you live in Philadelphia.  Counterexample: someone who lives in Chester. Therefore it’s false!  If a shape has four sides then it is a square.  Counterexample: a rectangle. Therefore it’s false!

Converses  The converse of a conditional switches the hypothesis (p) and the conclusion (q).  Its symbolic form is therefore q  p (read if q, then p).  It also may be true or false.

Example  All roses are flowers.  Conditional: If something is a rose, then it is a flower.  Converse: If something is a flower, then it is a rose.  Counterexample: A daisy!

Negation  Negation: The denial of a statement  Ex. The negation of An angle is obtuse is An angle is not obtuse.  If a statement is false, then its negation is true.

Inverse  Given a conditional statement, its inverse can be formed by negating both the hypothesis and conclusion.  The inverse of a true statement is not necessarily true.  ~p represents “not p” or the negation of p  The inverse of p  q is ~p  ~q

Example!  Write the inverse of the TRUE conditional Vertical angles are congruent. Determine if the inverse is true or false. If false, give a counterexample.  First, write the conditional in if-then form  What is the hypothesis?  What is the conditional?

Example continued..  The hypothesis is: two angles are vertical.  The conclusion is: the angles are congruent.  So the If-then form of the conditional is If two angles are vertical, then they are congruent.

 Now, negate both the hypothesis and the conclusion to form the inverse of the conditional.  Inverse: ????

Inverse  If two angles are NOT vertical, then they are NOT congruent.  Is the inverse TRUE or FALSE?  If false, how could we draw a counterexample to show that it is indeed false?

In the figure: A, B, and C are collinear. Points A, B, C, and D are in plane N.  A, B, and E lie in plane N.  True or False?  BC does not lie in plane N.  True or False?  A, B, C, and E are coplanar.  True or False?  A, B, and D are collinear.  True or False?

Identify the hypothesis and conclusion of each conditional statement  “If you don’t know where you are going, you will probably end up somewhere else.” (Laurence Peters, 1969)

Identify the hypothesis and conclusion of each conditional statement  If you are an NBA basketball player, then you are at least 5’2” tall.

Write each conditional in If-Then form.  “Happy people rarely correct their faults.” (La Rochefoucauld, 1678)  “A champion is afraid of losing.” (Billie Jean King, 1970s)  Adjacent angles have a common vertex.  Equiangular triangles are equilateral.

Write the negation of each statement.  A book is a mirror.

Write the negation of each statement.  Right angles are not acute angles.

Write the negation of each statement.  Rectangles are not squares.

Write the negation of each statement.  A cardinal is not a dog.

Write the negation of each statement.  You live in Dallas.

Write the converse, inverse, and contrapositive of each conditional.  All squares are quadrilaterals.  Determine if the converse, inverse, and contrapositive are true or false. If FALSE, give a counterexample.

Write the converse, inverse, and contrapositive of each conditional.  Three points not on the same line are noncollinear.  Determine if the converse, inverse, and contrapositive are true or false. If FALSE, give a counterexample.

Write the converse, inverse, and contrapositive of each conditional.  If a ray bisects an angle, then the two angles formed are congruent.  Determine if the converse, inverse, and contrapositive are true or false. If FALSE, give a counterexample.

Write the converse, inverse, and contrapositive of each conditional.  Acute angles have measures less than 90 o.  Determine if the converse, inverse, and contrapositive are true or false. If FALSE, give a counterexample.

Write the converse, inverse, and contrapositive of each conditional.  Vertical angles are congruent.  Determine if the converse, inverse, and contrapositive are true or false. If FALSE, give a counterexample.

In the figure: A, B, and C are collinear. Points A and X lie in plane M. Points B and Z lie in plane N. Determine whether each statement is true or false.  B lies in plane M.  A, B, and C lie in plane M.  A, B, X, and Z are coplanar.  BZ lies in plane N.

HOMEWORK – STUDY FOR QUIZ  WANT EXTRA CREDIT ON TOMORROW’S QUIZ?  MAKE FLASHCARDS FOR CONDITIONAL STATEMENT VOCABULARY  CONVERSE, INVERSE, CONTRAPOSITIVE, NEGATION, CONJECTURE, FALSE CONDITIONAL, HYPOTHESIS, CONCLUSION, COUNTEREXAMPLE

 Friday, February 20, 2015 Conditional Statement Quiz Today

Conditional Statement & Reasoning HW Review …. QUESTIONS?  Bellwork: PAIR – SHARE your homework with a partner before I collect it!

Conditional Statement Quiz – 30 minutes