Transcript Pre-AP Geometry 1

Pre-AP Geometry 1
Unit 2: Deductive Reasoning
Pre-AP Geometry 1 Unit 2
2.1 If-then statements, converse,
and biconditional statements
Conditional Statements
• Conditional Statement– A statement with two parts (hypothesis and conclusion)
– Also known as Conditionals
• If-then form
– A way of writing a conditional statement that clearly showcases
the hypothesis and conclusion p→q
• Hypothesis– The “if” part of a conditional statement
– Represented by the letter “p”
• Conclusion
– The “then” part of a conditional statement
– Represented by the letter “q”
Conditional Statements
• Examples of Conditional Statements
– If today is Saturday, then tomorrow is Sunday.
– If it’s a triangle, then it has a right angle.
– If x2 = 4, then x = 2.
– If you clean your room, then you can go to the
mall.
– If p, then q.
Conditional Statements
• Example 1
• Circle the hypothesis and underline the conclusion in
each conditional statement
– If you are in Geometry 1, then you will learn about the building
blocks of geometry
– If two points lie on the same line, then they are collinear
– If a figure is a plane, then it is defined by 3 distinct points
Conditional Statements
• Example 2
• Rewrite each statement in if…then form
– A line contains at least two points
a figure
a line, intersect
then it contains
at least two
– IfWhen
twoisplanes
their intersection
is points
a line
If two planes intersect, then their intersection is a
line.
– Two angles that add to 90° are complementary
If two angles add to equal 90°, then they are
complementary.
Conditional Statements
• Counterexample
– An example that proves that a given
statement is false
• Write a counterexample
– If x2 = 9, then x = 3
Conditional Statements
• Example 3
– Determine if the following statements are true
or false.
– If false, give a counterexample.
• If x + 1 = 0, then x = -1
• If a polygon has six sides, then it is a decagon.
• If the angles are a linear pair, then the sum of the
measure of the angles is 90º.
Conditional Statements
• Converse
– Formed by switching the if and the then part.
• Original
– If you like green, then you will love my new shirt.
• Converse
– If you love my new shirt, then you like green.
Biconditional Statements
• Can be rewritten with “If and only if”
• Only occurs when the statement and the
converse of the statement are both true.
• A biconditional can be split into a conditional and
its converse.
• p if and only if q
• All definitions can be written as biconditional
statements
Example
•
Give the converse of the statement.
– If the converse and the statement are both
true, then rewrite as a biconditional
statement
1. If it is Thanksgiving, then there is no school.
2. If an angle measures 90º, then it is a right
angle.
Quiz- Get out a piece of paper and
answer the following questions:
Underline the hypothesis and circle the
conclusion. Then, write the converse of the
statement. If the converse and the statement are
true, rewrite as a biconditional statement. If not,
give a counterexample.
1. If a number is divisible by 10, then it is
divisible by 5.
2. If today is Friday, then tomorrow is Saturday.
3. If segment DE is congruent to segment EF, then
E is the midpoint of segment DF.
Assignment
• Lesson 2.1
• P. 35 #2-30 even
Pre-AP Geometry 1 Unit 2
2.2: Properties from Algebra
p. 37
Properties of equality
• Addition property
– If a = b, then a + c = b + c
• Subtraction property
– If a = b, then a – c = b – c
• Multiplication property
– If a = b, then ac = bc
• Division property
– If a = b, then
a c  b c
Reasoning with
Properties from Algebra
• Reflexive property
– For any real number a, a = a
– EF  EF
• Symmetric property
– If a=b, then b = a
– If DE  FG, thenFG  DE
• Transitive Property
– If a = b and b = c, then a = c
– If ∠D  ∠E and ∠E  ∠F, then ∠D ∠F
• Substitution property
– If a = b, then a can be substituted for b in any equation or expression
• Distributive property
– 2(x + y) = 2x + 2y
Two-column proof
• A way of organizing a proof in which the
statements are made in the left column
and the reasons (justification) is in the
right column
• Given: Information that is given as fact in
the problem.
Reasoning with
Properties from Algebra
• Example 1
– Solve 6x – 5 = 2x + 3 and write a reason for each step
Statement
6x – 5 = 2x + 3
4x – 5 = 3
4x = 8
x=2
Reason
Given
Reasoning with
Properties from Algebra
Example 2
• 2(x – 3) = 6x + 6
•
•
•
•
1. Given
2.
3.
4.
5.
Reasoning with
Properties from Algebra
• Determine if the equations are valid or invalid,
and state which algebraic property is applied
– (x + 2)(x + 2) = x2 + 4
– x3x3 = x6
– -(x + y) = x – y
Warmup
• With a partner, Complete proof # 11 and 12 on
p. 40
Proving Theorems
Lesson 2.3
Pre-AP Geometry
Proofs
Geometric proof is deductive reasoning at work.
Throughout a deductive proof, the “statements”
that are made are specific examples of more
general situations, as is explained in the
"reasons" column.
Recall, a theorem is a statement that can be
proved.
Vocabulary
Definition of a Midpoint
The point that divides, or bisects, a segment into two
congruent segments.
If M is the midpoint of AB, then AM is congruent to MB
Bisect
To divide into two congruent parts.
Segment Bisector
A segment, line, or plane that intersects a segment at its
midpoint.
Midpoint Theorem
If M is the midpoint of AB, then AM = ½AB and MB = ½AB
Proof: Midpoint Formula
Given: M is the midpoint of Segment AB
Prove: AM = ½AB; MB = ½AB
Statement
1. M is the midpoints of segment AB
2. Segment AM= Segment MB,
or AM = MB
3. AM + MB = AB
4. AM + AM = AB, or 2AM = AB
5. AM = ½AB
6. MB = ½AB
Reason
1.
2.
3.
4.
Given
Definition of midpoint
Segment Addition Postulate
Substitution Property
(Steps 2 and 3)
5. Division Prop. of Equality
6. Substitution Property.
(Steps 2 and 5)
The Midpoint Formula
The Midpoint Formula
If A(x1, y1) and B(x2, y2) are points in a
coordinate plane, then the midpoint of segment
AB has coordinates:
 x  x y  y2 
M  1 2 , 1

2 
 2
x1  x2
Mx 
2
y1  y2
My 
2
The Midpoint Formula
Application:
Find the midpoint of the segment defined by the
points
A(5, 4) and B(-3, 2).
Midpoint Formula
Application:
Find the coordinates of the other endpoint
B(x, y) of a segment with endpoint C(3, 0)
and midpoint M(3, 4).
Vocabulary
Definition of an Angle Bisector
A ray that divides an angle into two adjacent
angles that are congruent.
If Ray BD bisects angle ABC, then ABD is
congruent to DBC
Angle Bisector Theorem
If BX is the bisector of ∠ABC, then the measure
of
∠ABX is one half the measure of ∠ABC
and the measure of ∠XBC one half of the
∠ABC.
A
X
B
C
Proof: Angle Bisector Theorem
Given: BX is the bisector of ∠ABC.
Prove: m ∠ABX = ½ m ∠ABC; m ∠XBC = ½m
∠ABC Statement
Reason
1. BX is the bisector of ∠ABC
1. Given
2. m∠ABX + m∠XBC =
m∠ABC
2. Angle addition postulate
3. m∠ ABX = m∠ XBC
3. Definition of bisector of an
angle
4. m∠ ABX + m∠ ABX =
4. Substitution property
m∠ ABC; 2 m∠ ABX = m∠ ABC
5. m∠ ABX = ½ m∠ ABC;
m∠ XBC = ½ m∠ ABC
5. Division property
Reasons used in proofs
1.
2.
3.
4.
Given
Definitions
Postulates
Theorems
Page 50
Pre-AP Geometry 1
• Complementary Angles
–Two angles that have a sum of 90º
–Each angle is a complement of the other.
Non-adjacent complementary
Adjacent
angles
complementary
angles
• Supplementary Angles
–Two angles that have a sum of 180º
–Each angle is a supplement of the other.
• Example 1
–Given that G is a supplement of H and
mG is 82°, find mH.
–Given that U is a complement of V, and
mU is 73°, find mV.
• Example 2
–T and S are supplementary.
The measure of T is half the measure of S.
Find mS.
• Example 3
–D and E are complements and D and F
are supplements. If mE is four times mD,
find the measure of each of the three angles.
• Vertical angles are congruent
–Given: angle 1 and angle 2 are vertical angles
3
–Prove∠1≅ ∠2
1
2
Statement
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
Find x and the measure of each angle.
∠A
32°
2x + 10
Page 56
Pre-AP Geometry 1
• Two lines that intersect to form right
angles
• We use the symbol ⊥ to show that lines
are perpendicular. LineC AB ⊥ Line CD
A
B
D
• Theorem 2-4: If two lines are perpendicular,
then they form congruent adjacent angles
• Theorem 2-5: If two lines form congruent
adjacent angles, then the lines are
perpendicular
• Theorem 2-6: If the exterior sides of two
adjacent angles are perpendicular, then the
angles are complementary.
Unit 2.6: Planning a proof
p. 60
Pre-AP Geometry 1
September 11, 2008
Parts of a proof
1. Statement of the theorem you are trying
to prove
2. A diagram to illustrate given information
3. A list of the given information
4. A list of what you are trying to prove
5. A series of Statements and Reasons that
lead from the given information to what
you are trying to prove.
Example proof of theorem 2-7
If 2 angles are supplements of congruent angles, then the two angles are
congruent.
Given: ∠2 ≅ ∠4
∠1 and ∠2 are supplementary
1
2
∠3 and ∠4 are supplementary
Prove: ∠1 ≅ ∠3
3
4
Statement
Reason
1. ∠1 and ∠2 are supplementary
∠3 and ∠4 are supplementary
1. Given
2. m ∠1 +m ∠2 =180; m ∠3 + m∠4
=180
2. Definition of supp. ∠’s
3. m ∠1 +m ∠2 = m ∠3 + m∠4
3. Substitution property
4. ∠2 ≅ ∠4
4.given
5. ∠1 ≅ ∠3
5. Subtraction property of equality
Theorem 2-8:
• If two angles are complements of
congruent angles, then the two angles are
congruent.
• Prove theorem 2-8. Use the proof from
theorem 2-7 (p. 61) to help. You may do
this with a partner. Due at end of hour.
Make sure you include all 5 parts (p. 60).