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

https://play.kahoot.it/#/k/e7e6a254-0ddb-4957-92bfb99911f8960f
 https://play.kahoot.it/#/k/e7e6a254-0ddb-4957-
92bf-b99911f8960f
Wednesday, February 18, 2015
 INDUCTIVE REASONING
Bell Ringer
 Find m<z
 Tell me WHY you could figure out what the
m<z was (What is the RELATIONSHIP?)
27o
27o
z
Inductive Reasoning and Conjecturing

Definitions
 Conjecture: An educated guess
 Inductive Reasoning: Looking at
several specific situations to arrive
at a conjecture
Example 1
 If you place the ball in one corner of a 10-
foot by 5-foot carom billiard table and
shoot it at a 45o 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 45o angle
from any corner of the table.
Some conjectures are:
 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.
Discussion
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.
 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.
Example 2
 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: ?
 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: ?
Example 3
 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.
 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.
Definition
 Counterexample: A false example
Example 4
 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.
Exit Ticket
 Explain the meaning

Determine if the conjecture if true or
false based on the given information.
Explain your answer.
 Describe why three

Given:
of conjecture in your
own words.
points on a circle
could never be
collinear.
 Explain how you can
prove that a
conjecture is false.
<1 and <2 are supplementary angles
<1 and <3 are supplementary angles.

Conjecture: <2 = <3
Thursday, February 19, 2015

Geometry - Lesson 2.2
TODAY’S OBJECTIVE:
SWBAT define conditional, the parts of a conditional,
converses and counterexamples.
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
90o.
 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