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Geometry Lesson 2.4
Reasoning with Properties
from Algebra
Warm-Up: Describing Solutions
Each equation below can be solved in one step
 Solve each one and write in words what you did
to solve it
 Example: 2x = 10
x = 5 (divided both sides by 2)
(a) 3x = 27 x = 9 (divided both sides by 3)

(b) x + 6 = –17
x = -23 (subtracted 6 from both sides)
(c) x – 9 = 18 x = 27 (added 9 to both sides)
(d) –x = 4
x = 4 (multiplied both sides by -1)
1. Algebraic Properties of Equality

Here are the formal properties that allow
you to solve algebraic equations…
The Property
1. Addition
If a = b, then a + c = b + c
2. Subtraction
If a = b, then a – c = b – c
3. Multiplication If a = b, then ac = bc
If a = b and c ≠ 0,
4. Division
then a ÷ c = b ÷ c
Property of Equality
Note: a, b, and c are real numbers
Algebraic Properties of Equality, cont.
Property of Equality
5. Reflexive
6. Symmetric
7. Transitive
The Property
For any real number, a = a
If a = b, then b = a
If a = b and b = c,
then a = c
If a = b, then a can be
8. Substitution
substituted for b
9. Distributive
a(b + c) = ab + ac
Note: a, b, and c are real numbers
2. Using Properties of Equality


You know how to solve equations, but can
you state why?
Example 1a: Solve 5x – 18 = 3x + 2 by
listing each step with a reason
Step in Solution
5x – 18 = 3x + 2
Reason
Given
2x – 18 = 2
if a = b, then a – c = b – c
2x = 20
if a = b, then a + c = b + c
x = 10
a b
If a = b, and c  0, then 
c c
Example 1b

Solve 55x – 3(9x + 12) = -64 and write
the REASON for each step
Step in Solution
55x – 3(9x + 12) = -64
55x – 27x – 36 = -64
28x – 36 = -64
28x = -28
x = -1
Reason
Given
a(b + c) = ab + ac
Combine like terms
if a = b, then a + c = b + c
a b
If a = b, and c  0, then 
c c
Practice 1a

List a reason for each step in the solution of
3x + 12 = 8x – 18
Step in Solution
3x + 12 = 8x – 18
Reason
Given
12 = 5x – 18
if a = b, then a – c = b – c
30 = 5x
if a = b, then a + c = b + c
x=6
a b
if a = b, and c  0, then 
c c
Practice 1b

Solve 4(2x + 5) = 2(x – 5) and list a reason
for each step
Step in Solution
4(2x + 5) = 2(x – 5)
8x + 20 = 2x – 10
Reason
Given
a(b + c) = ac + ac
6x + 20 = –10
if a = b, then a – c = b – c
6x = –30
if a = b, then a – c = b – c
x = –5
a b
if a = b, and c  0, then 
c c
3. Properties of Length and Angle Measures
ANGLE
MEASURES
For any A,
For any AB,
Reflexive
AB = AB
mA = mA
If AB = CD,
If mA = mB,
Symmetric
then CD = AB then mB = mA
If AB = CD
If mA = mB
Transitive and CD = EF, and mB = mC,
then AB = EF then mA = mC
Property of
Equality
SEGMENT
LENGTHS
Example 2a: Segment Lengths


In the diagram, AB = CD (equal lengths)
Show that AC = BD
A B
C D
Step
AB = CD
AC = (AB + BC)
BD = (CD + BC)
(AB + BC) = (CD + BC)
AC
=
BD
Reason
Given
if B between A&C, then
AC = AB + BC
if C between B&D, then
BD = CD + BC
if AB = CD, then
AB + BC = CD + BC
if a = b, then subst a for b
Example 2b: Angle Measures


In the diagram, mBEA = mCED
Show that mAEC = mBED
Step
Reason
mBEA = mCED Given
if B on interior, then
mAEC = mBEA + mBEC mBEA + mBEC =
mAEC
mBED = mCED + mBEC Same for C on interior
mBEA + mBEC if a = b, then
= mCED + mBEC a + c = b + c
if a = b, then
mAEC = mBED
subst a for b
Practice 2a
Step
WY = XZ
WY = WX + XY
XZ = YZ + XY
Reason
Given
if X between W&Y, then
WY = WX + XY
if Y between X&Z, then
XZ = YZ + XY
WX + XY = YZ + XY
if a = b, then subst a for b
WX = YZ
if a = b, then a – c = b – c
Practice 2b


In the diagram, mAEC = mDEB
Show that mAEB = mCED
Step
Reason
mAEC = mDEB Given
if B on interior, then
mAEC = mAEB + mBEC
mAEB + mBEC = mAEC
mDEB = mCED + mBEC Same for C on interior
mAEB + mBEC if a = b, then
= mCED + mBEC subst a for b
if a = b, then
mAEB = mCED
a–c=b–c
Assignment
Ch. 2.4 (Pg. 99-101)
#10-26 EVEN, #32