Download 2-6 Geometric Proofs - Western High School

2-1 Using Inductive Reasoning to Make Conjectures
Inductive Reasoning – Based on patterns or past performance.
Conjecture – Statement you believe to be true.
Exercise A – Find the next item in each pattern, or make a conjecture.
1)
January, April, July, … October
2)
7, 21, 35, 49, … 63
3) ….
20, 10, 5, … 2.5
4)
Example B – Complete the conjecture.
1) The sum of two positive numbers is
. positive
2) The product of two odd numbers is
. odd
3) Make a conjecture about the lengths of male and female whales based on the data.
Female whales are longer than male whales.
Counterexample –
Example that proves a conjecture false.
Example C – Show that the conjecture is false by finding a counterexample.
1) For all integers n, n3 > 0
n=0
03 > 0 FALSE
2) Two complementary angles are not congruent.
TWO 45 DEGREE ANGLES.
FALSE
3) The monthly high temperature in Abilene is never below 90°F for two months in a row.
Jan. & Feb.
4) For any real number x, x2 ≥ x.
.12 > .1
.01> .1 False
x = .1
5) For any three points in a plane, there are three different lines that contain two of the points.
False. 3 collinear points.
2-2 Conditional Statements
Conditional statement – “If, then” statement format
Hypothesis – “p”
everything that follows “If” and ends at before “then” in a conditional statement.
Conclusion – “q”
everything that follows “then” in a conditional statement.
Example A: Identify the hypothesis and conclusion of each statement. Do not include “if” & “then”
1) If it is Saturday, then there is no school.
2) If a polygon has four right angles, then the polygon is a square.
3) A number is a rational number if it is an integer.
Example B: Write each statement in if-then form.
1) An obtuse triangle has exactly one obtuse angle.
If a triangle is obtuse, then it has exactly one obtuse angle.
2) The midpoint M of a segment bisects the segment.
If M is the midpoint of a segment, then it bisects the segment.
3) Two angles that are complementary are acute.
If two angles are complementary, then they are acute.
Truth Value – evaluates conditional statement to be “TRUE” or “FALSE”
Example C: Determine if each conditional statement is true. If false, give a counter example.
1) If today is Sunday, then tomorrow is Monday. True
2) If an angle is obtuse, then it has a measure of 100 degrees. False 105
3) If an odd number is divisible by 2, then 8 is a perfect square. True
4) If two angles are acute, then they are congruent. False 60 - 50
Negation – “NOT”
Related Conditionals
Definitions
Given
(conditional)
p
Converse
q
Inverse
~p
Example
q
If it is raining, then the home
team wins.
If the home team wins, then
it is raining.
p
~q
If it is NOT raining, then then
the home team will NOT win.
If not p, then not q
~ q
~p
Contrapositive
If not q, then not p
If the home team does NOT
win, then it is NOT raining.
Example D: Write the converse, inverse, and contrapositive of the conditional statement.
Determine if each is true or false. If false, give a counterexample.
Conditional
If an angle is obtuse, then it has a
measure of 100°.
Converse
If an angle has a measure of 100°,
then then it is obtuse.
True
If an angle is NOT obtuse, then it is
NOT 100°.
True
Inverse
False – 110 degrees
If an angle has a measure is NOT
Contrapositive 100°, then then it is NOT obtuse.
False – 110 degrees
Example E: Write the converse, inverse, and contrapositive of the conditional statement.
Determine if each is true or false. If false, give a counterexample.
Conditional
If freezing rain is falling, then the air
temperature is 32 or less.
True
False, Clear and cold.
Converse
If the air temperature is 32 or less,
then freezing rain is falling.
False, Clear and cold.
Inverse
If freezing rain is NOT falling, then
the air temperature is NOT 32 or
less.
If the air temperature is NOT 32 or
True
Contrapositive less, then freezing rain is NOT
falling.
2-3 Using Deductive Reasoning to Verify Conjectures
Deductive Reasoning – based on logic, facts, definitions, theorems & scientific
methods
Example A – Is each conclusion a result of inductive or deductive reasoning?
1) There is a myth that toilets and sinks drain in opposite directions in the Southern and
Northern Hemispheres. However, if you were to observe sinks draining in two hemispheres,
you would see that this myth is false.
INDUCTIVE
2) There is a myth that you should not touch a baby bird that has fallen from its nest because
the mother bird will disown the baby if she detects human scent. However, biologists have
shown that birds cannot detect human scent. Therefore, the myth cannot be true.
DEDUCTIVE
3) There is a myth that an eelskin wallet will demagnetize credit cards because the skin of the
electric eels used to make the wallet holds an electric charge. However, eelskin products
are not made from electric eels. Therefore, the myth cannot be true.
DEDUCTIVE
4) There is a myth that you can balance an egg on its end only on the spring equinox. A
person was able to balance an egg on July 8th, September 21st, and December 19th.
Therefore, this myth is false.
INDUCTIVE
Example B – Draw a conclusion from the given information.
1) Given: If a team wins 10 games, then they play in the finals. If a team plays in the finals,
then they travel to Boston. The Ravens won 10 games.
Conclusion: The Ravens travel to Boston.
2) Given: If two angles form a linear pair, then they are adjacent. If two angles are adjacent,
then they share a side.
and
form a linear pair.
Conclusion: .
and
share a side.
3) Given: If a polygon is a triangle, then it has three sides. If a polygon has three sides, then it
is not a quadrilateral. Polygon P is a triangle.
Conclusion: Polygon P is not a quadrilateral.
2-4 Biconditional Statements and Definitions
Biconditional Statements – written in the “p if and only if q” format
Example A: Write the conditional statement and converse within each biconditional.
1) Two angles are congruent if and only if their measures are equal.
Conditional: If two angles are congruent, then their measures are equal.
Converse: If two angles are equal, then they are congruent.
2) A solution is a base ↔ it has a pH greater than 7.
Conditional: If a solution is a base, then it has a pH greater than 7.
Converse: If a solution it has a pH greater than 7, then it is a base
Example B: For each conditional statement, write the converse and a biconditional statement.
p
q
1) If
, then
.
Converse: If x=3, then 2x +5 = 11
Biconditional: 2x +5 = 11 IFF x = 3
2) If a point is a midpoint, then it divides the segment into two congruent segments.
Converse: If a point divides the segment into two congruent segments, then it is a midpoint.
Biconditional: A point is a midpoint IFF the segment into two congruent segments.
Example C: Determine if the biconditional is true. If false, give a counterexample.
1) A square has a side length of 5 if and only if it has an area of 25. TRUE
2) If y=5, then y2=25. FALSE. Y = -5
In geometry, biconditional statements are used to write definitions. The definitions you will learn
throughout Geometry are biconditional. You MUST remember that the conditional and converse
of DEFINITIONS are always true!
Example D: Write each definition as a biconditional.
1) A triangle is a three sided polygon. A polygon is a triangle IFF it has three sides.
2) A right angle measures 90 . An angle is a right triangle IFF it has a measure of 90 degrees.
2-5 Algebra Proofs/Intro to Geometric Proofs
Proof – Arguments that use logic, definitions, properties, & postulates.
Complete each proof.
• Each 2-column proof has 2 columns.
• The left column has statements and the right columns has reasons.
• The statement column contains equations and the reason column contains
definitions, properties, & postulates and theorems.
• The first reason is ALWAYS given.
Example A
=9
1) Given:
Prove: x = 3
Statements
1.
=9
Reasons
1. given
2. 4x + 6 =18
2. Multiplication property of equality
3. 4x =12
3. Subtraction property of equality
4.
4. Division property of equality
X=3
2) Given: 3(x - 2) + 5 = 47
Statements
Prove: x=16
Reasons
1. 3(x - 2) + 5 = 47
1. Given
2. 3x -6 + 5 = 47
2. Distributive Property
3. 3x – 1 =47
3. Simplify
4. 3x = 48
4. Addition Prop. of
equality
5. x=16
5. Division Prop. of
equality
Some Algebraic
Properties work
for Geometric
Properties!
REMEMBER
THESE!!!
Example B –
Identify the
property that justifies each statement.
1) If m<1 = m< 2, then m<2 = m< 1. Symmetric Prop.
2) If m<1 = 90 and m<2= m <1, then m<2= 90 . Transitive Prop.
3) If AB=RS and RS=WY, then AB=WY. Transitive Prop.
4) If AB=CD, then 1 AB = 1 CD Multiplication Prop. of Equality
2
2
5) If m<1+ m< 2= 110 and m<2= m<3, then m<1 + m<3=110. Substitution Prop.
6) RS=RS. Reflexive Prop.
7) If AB=RS, then AB + 9 = RS + 9 Addition Prop. of Equality
8) If m<1= m<2 and m<2= m<3, then m<1= m< 3. Transitive Prop.
2-6 Geometric Proofs
Remember: A theorem is anything that can be proved.
•
Linear Pair Theorem o
•
Congruent Supplements Theorem
o If two angles are supplementary
If two angles form a linear pair; then they are supplementary.
to the same angle (or to two congruent angles), then
the two angles are congruent.
Congruent Complements Theorem
o If two angles are complementary
to the same angle (or to two congruent angles), then
the two angles are congruent.
Right Angle Congruence Theorem
o All right angles are congruent.
A)
C)
1. <1=<2
1. Given
1. ABC is a right
angle.
1. Given
2. m<1= m<2
2. Def. of congruent
angles
2. m<ABC=90°
2. Def of Right
angles
B)
1. m<1= m<2
2.
<1=<2
E)
D)
1. Given
2. . Def. of congruent
angles
2. m<1+m<2 = 180
1. Given
2. Def. of Supplementary
angles
Given:<1 and <2 are supplementary, and <1 = <3
Prove: <3 and <2 are supplementary
Statements
1. <1 & <2 are Supplementary. <1= <3
2. m<1 + m<2 = 180
3. m<1=m<3
4. m<3 + m<2=180
5.
1. <1 and <2 are
supplementary
<3 and <2 are supplementary
1.
2.
3.
4.
Reasons
Given
Def. of Supplementary angles
Def. of Congruent angles
Substitution Prop.
5. Def. of Supplementary angles
F)
S
Given: Q is between P and R, R is between Q and S, and PR=QS.
Prove: PQ=RS
R
Statements
Reasons
1. Q is between P and R, R is between Q and S, PR=QS.
1. Given
2.
PQ + QR =PR QR + RS = QS
2. Segment Add, Post.
3.
PQ + QR = QR + RS
3. Substitution Prop
4.
G)
Q
P
4. Subtraction Prop. of EQ.
PQ=RS
Given: <BAC is a right angle; <2=<3
Prove: <1 and <3 are complementary
C
Statements
Reasons
1. <BAC is a right angle; <2=<3
1. Given
2. m<BAC=90
2. Def. of Right angles.
3. m<1 +m<2 = m<BAC
3. Angle Add. Post.
4. m<1 +m<2 = 90
4. Substitution
5. m<2=m<3
5. Def. of congruent angles
6. m<1 +m<3= 90
6. Substitution
7. <1 and <3 are complementary
7. Def. of complementary angles
H)
Given: Q is the midpoint of PR. R is the midpoint of QS.
Prove: PR=QS
S
P
Statements
1. Q is midpoint of PR. R is midpoint of QS .
2. QR=RS
3. PQ=RS
4. PQ + QR= RS + QR
5. PQ +QR=PR, RS+QR=QS
6. PR=QS
Q
R
Reasons
1. Given
2. Def. of Midpoint
3. Transitive Prop.
4. Addition Prop of Equality
5. Segment Addition Postulate
6. Substitution
I)
Given: X is the midpoint of WY .
Prove: WX+YZ=XZ
Statements
1. X is midpoint of WY.
2. WX=XY
3. XY + YZ =XZ
4. WX + YZ =XZ
J)
W
X
Y
Z
Reasons
1. Given
2. Def. of Midpoint
3. Segment Addition Property
4. Substitution
Given: AB= DE, BC =EF
Prove: AC =DF
C
B
F
E
A
Statements
Reasons
D
1.
2.
3.
4.
5.
___ ___ ___ ___
AB= DE, BC =EF
AB= DE, BC =EF
AB + BC = DE + EF
AB + BC=AC, DE + EF=DF
AC = DF
__ __
1. Given
2. Def. of Congruent Segments
3. Substitution Property
4. Segment Addition Postulate
5. Substitution Property
6.
AC =DF
6. Def. of Congruent Segments
K)
Given: BC=DE
Prove: AB+DE=AC
C
A
B
Statements
E
D
Reasons
1. BC=DE
1. Given
2. AB+BC=AC
2. Segment Add. Postulate
3. AB+DE=AC
3. Substitution
L)
Given: PR QS
Prove: PQ RS
P
Statements
Reasons
1.
1. PR QS
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
M)
R
Q
Given: 1 4.
Prove: 2 3
1
Statements
2
3
4
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
8.
8.
9.
9.
S