Transcript 2.5 Proving Statements About Segments

Reasoning with Properties
of Algebra &
Proving Statements About
Segments
CCSS: G-CO.12
CCSS:G-CO.12
• Make formal geometric constructions with a
variety of tools and methods (compass and
straightedge, string, reflective devices, paper
folding, dynamic geometric software, etc.).
Copying a segment; copying an angle; bisecting
a segment; bisecting an angle; constructing
perpendicular lines, including the perpendicular
bisector of a line segment; and constructing a
line parallel to a given line through a point not on
the line.
Essential Question(s)
• What algebra properties apply to angles
and segments?
• How do we use properties of length and
measure to justify segment and angle
relationships?
• How do we justify statements about
congruent segments?
Activator:
• Work with your partner. Make a list of
Properties of Equality for Algebra. Give
examples for each property. Solve writing
down your reasoning for each step:
6x + 3 = 9(x -1).After you finish walk
around to compare your results with the
other groups.
Activator:
• Given: AB = BC
• Prove: AC = 2(BC)
A
B
C
Objectives
Review properties of equality and use
them to write algebraic proofs.
Identify properties of equality and
congruence.
• In Geometry you accept postulates &
properties as true.
• You use Deductive Reasoning to prove
other statements.
• In Algebra you accept the Properties of
Equality as true also.
Algebra 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 a • c = b • c
Division Property:
If a = b, then a/c = b/c (c ≠ 0)
More Algebra Properties
•
•
•
•
•
•
Reflexive Property:
a = a (A number is equal to itself)
Symmetric Property:
If a = b, then b = a
Transitive Property:
If a = b & b = c, then a =c
2 more Algebra Properties
• Substitution Properties: (Subs.)
• If a = b, then “b” can replace “a” anywhere
• Distributive Properties:
• a(b +c) = ab + ac
A proof is an argument that uses logic, definitions,
properties, and previously proven statements to show
that a conclusion is true.
An important part of writing a proof is giving
justifications to show that every step is valid.
Example 1: Algebra Proof
3x + 5 = 20
-5 -5
3x = 15
3
3
x=5
5=x
1. Given Statement
2. Subtr. Prop
3. Division Prop
4. Symmetric Prop
Example 2 :
 Addition Proof
A
Given: mAOC = 139
Prove: x = 43
Statements
1. mAOC = 139, mAOB = x,
mBOC = 2x + 10
2. mAOC = mAOB + mBOC
3.
4.
5.
6.
7.
139 = x + 2x + 10
139 = 3x + 10
129 = 3x
43 = x
x = 43
B
x
(2x +
O 10)
1.
2.
3.
4.
5.
6.
7.
C
Reasons
Given
 Addition Prop.
Subs. Prop.
Addition Prop
Subtr. Prop.
Division Prop.
Symmetric Prop.
Example 3: Segment Addition Proof
Given: AB = 4 + 2x
A
BC = 15 – x
4 + 2x
AC = 21
Prove: x = 2
1.
2.
3.
4.
5.
6.
Statements
AB=4+2x, BC=15 –
x, AC=21
AC = AB + BC
21 = 4 + 2x + 15 – x
21 = 19 + x
2=x
x=2
B
15 – x
C
Reasons
1. Given
2. Segment Add. Prop.
3. Subst. Prop.
4. Combined Like
Term.
5. Subtr. Prop.
6. Symmetric Prop.
You learned in Chapter 1 that segments with
equal lengths are congruent and that angles with
equal measures are congruent. So the Reflexive,
Symmetric, and Transitive Properties of Equality
have corresponding properties of congruence.
Theorem
• A true statement that follows as a result of
other true statements.
• All theorems MUST be proved!
2-Column Proof
• Numbered statements and corresponding
reasons in a logical order organized into 2
columns.
statements
reasons
1.
1.
2.
2.
3.
3.
etc.
Geometry Properties of
Congruence
1. Reflexive Property: AB  AB
A  A
2. Symmetric Prop: If AB  CD, then CD 
AB
If A  B, then B 
A
3. Transitive Prop:
If AB  CD and CD  EF, then AB  EF
IF A  B and B  C, then A 
Theorem 2.1- Properties of Segment
Congruence
• Segment congruence is reflexive,
symmetric, & transitive.
For any AB, AB  AB.
If AB  BC and BC  CD, then AB  CD.
If AB  BC, then BC  AB.
Proof of symmetric part of thm.
2.1
Statements
1. AB  BC
2. AB = BC
3. BC = AB
4. BC  AB
Reasons
1.
2.
3.
4.
Given
Defn. of congruent segs.
Symmetric prop of =
Defn. of congruent segs.
Paragraph Proof
• Same argument as a 2-column proof, but
each step is written as a sentence; therefore
forming a paragraph.
P
X
Y
Q
• You are given that line segment PQ is congruent
with line segment XY. By the definition of congruent
segments, PQ=XY. By the symmetric property of
equality XY = PQ. Therefore, by the definition of
congruent segments, it follows that line segment XY
congruent to line segment PQ.
Ex: Given: PQ=2x+5
QR=6x-15
PR=46
Prove: x=7
1.
2.
3.
4.
5.
6.
Statements
PQ=2x+5, QR=6x-15,
PR=46.
PQ+QR=PR
2x+5+6x-15=46
8x-10=46
8x=56
x=7
Reasons
1. Given
2.
3.
4.
5.
6.
Seg + Post.
Subst. prop of =
Simplify
+ prop of =
Division prop of =
Ex: Given: Q is the midpoint of PR.
PR
Prove: PQ and QR =
2
1.
2.
3.
4.
5.
6.
Statements
Q is midpt of PR
PQ=QR
PQ+QR=PR
QR+QR=PR
2QR=PR
QR= PR
2
PR
7. PQ=
2
1.
2.
3.
4.
5.
6.
Reasons
Given
Defn. of midpt
Seg + post
Subst. prop of =
Simplify
Division prop of =
7. Subst. prop
What did I learn Today?
• Name the property for each of the
following steps.
• P  Q, then Q  P
Symmetric Prop
• TU  XY and XY  AB, then TU  AB
Transitive Prop
• DF  DF
Reflexive