Download 2-4 to 2

2.4 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
AB ≅ AB
∠A≅∠A
 Symmetric Property
If AB ≅ CD, then CD ≅ AB
If ∠ A ≅ ∠ B, then ∠ B ≅ ∠ A
 Transitive Property
If AB ≅ CD and CD ≅ EF, then AB ≅ EF
If ∠ A ≅ ∠ B and ∠ B ≅ ∠ C,then ∠ A ≅ ∠ C

(mirror)
(twins)
(triplets)
Ex: Solve the equation &
write a reason for each step.
1.
2(3x+1) = 5x+14
1.
Given
2.
6x+2 = 5x+14
2.
Distributive prop
3.
x+2 = 14
3.
Subtraction prop of =
4.
x = 12
4.
Subtraction prop of =
Solve 55z-3(9z+12) = -64 &
write a reason for each step.
1.
55z-3(9z+12) = -64
1.
Given
2.
55z-27z-36 = -64
2.
Distributive prop
3.
28z-36 = -64
3.
Simplify (or collect like terms)
4.
28z = -28
4.
Addition prop of =
5.
z = -1
5.
Division prop of =
Solving an Equation in Geometry with justifications
NO = NM + MO
Segment Addition Post.
4x – 4 = 2x + (3x – 9) Substitution Property of Equality
4x – 4 = 5x – 9
–4 = x – 9
5=x
Simplify.
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)°
8x = 9x – 11
 Add. Post.
Subst. Prop. of Equality
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…
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
2.5 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
b.
Supplementary
c.
Vertical
d.
Adjacent
1
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
Theorem 2-5 If
right angles are congruent
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:
<1 and <2 are supplementary
<3 and <2 are supplementary
Prove: <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 __________.
CLASSWORK

Page 118-119

#24-35