2.5 – Reasoning Using Properties of Algebra
2.6 – Prove Statements about Segments and Angles
Name
Addition Property
Property of Equality
If a = b, then a + c = b + c
Subtraction Property
If a = b, then a – c = b – c
Multiplication
Property
Division Property
If a = b, then ac = bc
Distribution
Property
Substitution
Property
If a = b, then a/c = b/c
If a(x + b), then ax + ab
If a = b, then b can be substituted in for a
Reflexive Property
For any real #, a = a
Symmetric Property
If a = b, then b = a
Transitive Property
If a = b and b = c, then a = c
Name
Property of Congruence
Reflexive
Any geometric object is
congruent to itself.
AB  AB
Symmetric
If object a  b, then object b  a
Transitive
If object a  b and object b  c,
then object a  c
1. Use the property to complete the statement.
a. Multiplication Property of Equality:
3m2
If m1 = m2, then 3m1 = ____________.
b. Substitution Property of Equality:
5(20) = 100
If a = 20, then 5a = __________.
Statement
If 4(x + 7), then 4x + 28
BD  BD
Reason
Distributive Prop
Reflexive Prop
If 2x + 5 = 9, then 2x = 4
Subtraction Prop
If A  B and B  C , then A  C
Transitive Prop
If x – 7 = 2, then x = 9
Addition Prop
If XY  AB, then AB  XY
If 4x = 12, then x = 3
Symmetric Prop
Division Prop
PROOFS!
All proofs start and end the same. It is very
helpful to have a picture to refer to. We are
given information and then are told to prove
something.
Format of Proofs
Given:
Prove:
Picture
Statements
Reasons
1. What you are given 1. Given
2.
2.
3.
3.
… To Prove
…
Steps for Proofs:
1. Start with given and mark pictures
2. What do the givens tell us?
3. What else do we know by looking at the
picture?
4. End the proof with what we are trying to prove
Complete the logical argument by writing a reason for each step.
Statements
1.
8x – 34 = 6
Reasons
1.
given
__________________
2.
8x = 40
2.
Addition Prop
__________________
3.
x=5
3.
Division Prop
__________________
Complete the logical argument by writing a reason for each step.
Statements
Reasons
1.
4x – 7 = 6x + 7
1.
given
__________________
2.
-2x – 7 = 7
2.
__________________
Subtraction Prop
3.
-2x = 14
3.
Addition Prop
__________________
4.
x = -7
4.
Division Prop
__________________
Complete the logical argument by writing a reason for each step.
Statements
Reasons
1.
5(x – 3) = 4(x + 2)
1.
given
___________________
2.
5x – 15 = 4x + 8
2.
Distributive Prop
__________________
3.
4.
x – 15 = 8
x = 23
3. ___________________
Subtraction Prop
4.
Addition Prop
___________________
11. Solve the equation. Write a reason for each step
4x – 9 = 2x + 11
Statements
1.
4x – 9 = 2x + 11
Reasons
1.
Given
2. 2x – 9 = 11
2.
Subtraction Prop.
3.
2x = 20
3.
Addition Prop.
4.
x = 10
4.
Division Prop.
12. Solve the equation for y. Write a reason for each step.
12x – 3y = 30
Statements
Reasons
1. 12x – 3y = 30
1. Given
2. –3y = -12x + 30
2. Subtraction Prop.
3.
y = 4x – 10
3. Division Prop.
13. Complete the proof by writing a reason for each step.
GIVEN: AL = SK
PROVE: AS = LK
Statements
Reasons
1.
AL = SK
1.
C. given
___________________
2.
LS = LS
2.
E. Reflexive Prop
___________________
3. AL + LS = SK + LS
3.
F. Addition Prop
____________________
4.
AL + LS = AS
Segment Addition Post.
4. A.
____________________
5.
SK + LS = LK
Segment Addition Post.
5. D.
____________________
6.
AS = LK
6.
B. Substitution Prop
____________________
14. Complete the proof.
GIVEN: M is the midpoint of AB , AM = 6in
PROVE: AB = 12in
Statements
Reasons
1.
M is the midpoint of AB
given
1. ___________________
2.
AM = MB
2.
Def. of midpt
___________________
3.
AM = 6in
3.
given
___________________
4.
AM + MB = AB
4.
Segment Add. Post.
___________________
5.
AM + AM = AB
Substitution
5. ___________________
6.
6 + 6 = AB
6. ___________________
Substitution
7.
AB = 12in
7.
___________________
Simplify
15. Complete the proof.
GIVEN: AB bisects CD at O, CO = 9cm
PROVE: CD = 18in
Statements
Reasons
1.
AB bisects CD at O
1.
given
_______________________
2.
CO = OD
2.
Def. of segment bisector
_______________________
3.
CO = 9cm
given
3. _______________________
4.
CO + OD = CD
4.
Segment Add. Post.
_______________________
5.
CO + CO = CD
5.
Substitution
_______________________
6.
9 + 9 = CD
6.
Substitution
_______________________
7.
CD = 18cm
7.
Simplify
_______________________
16. Complete the proof.
GIVEN: BD bisects ABC, mABD = 20°
PROVE: mABC = 40°
Statements
Reasons
1. BD bisects ABC
1. Given
2. mABD = mDBC
2. Def. of angle bisector
3. mABD = 20°
3. given
4. mABD + mDBC = mABC
4. Angle Addition Prop.
5. mABD + mABD = mABC
5. Substitution
6. 20° + 20° = mABC
6. Substitution
7. mABC = 40°
7. Simplify
HW Problem
2.5
2.6
108-111
116-119
1, 3-5, 7, 9, 17, 21-26, 41, 42
1-12, 17, 18, 21, 22, 24
2.6 # 18
D
D
1.
2.
3.
4.
5.
6.
mABC = 90°
mABD + mDBC = 90°
6x + 3x – 9 = 90
9x – 9 = 90
9x = 99
x = 11
1. Given
2. Angle Addition Postulate
3. Substitution
4. Simplify
5. Addition
6. Division