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Transcript
2.5 Reasoning with
Properties from Algebra
?
What are we doing, &
Why are we doing this?
• In algebra, you did things because you
were told to….
• In geometry, we can only do what we
can PROVE…
• We will start by justifying algebra steps
(because we already know how)
• Then we will continue justifying steps
into geometry…
But first…we need to
1. Learn the different properties / justifications
2. Know format for proving / justifying mathematical
statements
3. Apply geometry properties to proofs
Properties of Equality
(from algebra)
Addition property of equality- if a=b, then a+c=b+c.
(can add the same #, c, to both sides of an equation)
Subtraction property of equality - If a=b, then a-c=b-c.
(can subtract the same #, c, from both sides of an equation)
Multiplication prop. of equality- if a=b, then ac=bc.
Division prop. of equality- if a=b, then
a
b

c
c
Properties of Equality (Algebra)
Reflexive prop. of equality- a=a
Symmetric prop of equality- if a=b, then b=a.
Transitive prop of equality- if a=b and b=c, then a=c.
Substitution prop of equality- if a=b, then a can be plugged in for b and
vice versa.
Distributive prop.- a(b+c)=ab+ac
OR
(b+c)a=ba+ca
Properties of Equality (geometry)
Reflexive Property
(mirror)
AB ≅ AB
∠A≅∠A
Symmetric Property (twins)
If AB ≅ CD, then CD ≅ AB
If ∠ A ≅ ∠ B, then ∠ B ≅ ∠ A
Transitive Property (triplets)
If AB ≅ CD and CD ≅ EF, then AB ≅ EF
If ∠ A ≅ ∠ B and ∠ B ≅ ∠ C, then ∠ A ≅ ∠ C
Ex: Solve the equation & write a
reason for each step.
1. 2(3x+1) = 5x+14
2. 6x+2 = 5x+14
1. Given
2. Distributive prop
3. x+2 = 14
4. x = 12
3. Subtraction prop of =
4. Subtraction prop of =
Solve 55z-3(9z+12) = -64 & write
a reason for each step.
1.
2.
3.
4.
5.
55z-3(9z+12) = -64
55z-27z-36 = -64
28z-36 = -64
28z = -28
z = -1
1. Given
2. Distributive prop
3. Simplify (or collect like
terms)
4. Addition prop of =
5. Division prop of =
Solving an Equation in Geometry with Justifications
NO = NM + MO
4x – 4 = 2x + (3x – 9)
Substitution Property of Equality
4x – 4 = 5x – 9
Simplify.
–4 = x – 9
5=x
Segment Addition Post.
Subtraction Property of Equality
Addition Property of Equality
Solve, Write a
justification
for each step.
mABC = mABD + mDBC
8x° = (3x + 5)° + (6x – 16)°
 Add. Post.
Subst. Prop. of Equality
8x = 9x – 11
Simplify.
–x = –11
Subtr. Prop. of Equality.
x = 11
Mult. Prop. of Equality.
Remember!
Numbers are equal (=) and
figures are congruent ().
Identifying Property of Equality and
Congruence
Identify the property that justifies each
statement.
A. QRS  QRS
Reflex. Prop. of .
B. m1 = m2 so m2 = m1
Symm. Prop. of =
C. AB  CD and CD  EF, so AB  EF. Trans. Prop of 
D. 32° = 32°
Reflex. Prop. of =
Example
from
scratch…
2.6 Proving Angles
Congruent
Proving Angles Congruent
• Vertical Angles: Two angles whose sides
form two pairs of opposite rays; form two
pairs of congruent angles
<1 and <3 are
Vertical angles
1
4
<2 and <4 are
Vertical angles
2
3
Proving Angles Congruent
• Adjacent Angles: Two coplanar angles that
share a side and a vertex
1
2
<1 and <2 are
Adjacent Angles
1
2
Proving Angles Congruent
• Complementary Angles: Two angles whose
measures have a sum of 90°
50°
2
40°
1
• Supplementary Angles: Two angles whose
measures have a sum of 180°
105°
3
4
75°
Identifying Angle Pairs
In the diagram identify pairs of numbered
angles that are related as follows:
a. Complementary
1
b. Supplementary
c. Vertical
d. Adjacent
5
2
4
3
Making Conclusions
Whether you draw a diagram or use a given
diagram, you can make some conclusions
directly from the diagrams. You CAN conclude
that angles are
• Adjacent angles
• Adjacent supplementary angles
• Vertical angles
Making Conclusions
Unless there are markings that give this
information, you CANNOT assume
• Angles or segments are congruent
• An angle is a right angle
• Lines are parallel or perpendicular
Theorems About Angles
Theorem 2-1 Vertical Angles Theorem
Vertical Angles are Congruent
Theorem 2-2 Congruent Supplements
If two angles are supplements of the same
angle or congruent angles, then the two angles
are congruent
Theorems About Angles
Theorem 2-3 Congruent Complements
If two angles are complements of the same
angle or congruent angles, then the two angles
are congruent
Theorem 2-4 All right angles are congruent
Theorem 2-5 If two angles are congruent and
supplementary, each is a right angle
Proving Theorems
Paragraph Proof: Written as sentences in a
paragraph
Given: <1 and <2 are
vertical angles
Prove: <1 = <2
1
3
2
Paragraph Proof: By the Angle Addition Postulate, m<1
+ m<3 = 180 and m<2 + m<3 = 180. By substitution,
m<1 + m<3 = m<2 + m<3. Subtract m<3 from each
side. You get m<1 = m<2, which is what you are trying
to prove.
Proving Theorems
Given:
Prove:
<1 and <2 are supplementary
<3 and <2 are supplementary
<1 = <3
Proof: By the definition of supplementary
angles, m<___ + m<____ = _____ and m<___ +
m<___ = ____. By substitution, m<___ +
m<___ = m<___ + m<___. Subtract m<2 from
each side. You get __________.