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Name ________________________________________ Date __________________ Class__________________
Reteach
LESSON
2-5
Algebraic Proof
A proof is a logical argument that shows a conclusion is true. An algebraic proof uses
algebraic properties, including the Distributive Property and the properties of equality.
Properties of
Equality
Symbols
Examples
Addition
If a = b, then a + c = b + c.
If x = − 4, then x + 4 = − 4 + 4.
Subtraction
If a = b, then a − c = b − c.
If r + 1 = 7, then r + 1 − 1 = 7 − 1.
Multiplication
If a = b, then ac = bc.
If
Division
If a = 2 and c ≠ 0, then
Reflexive
a=a
15 = 15
Symmetric
If a = b, then b = a.
If n = 2, then 2 = n.
Transitive
If a = b and b = c, then a = c.
If y = 32 and 32 = 9, then y = 9.
Substitution
If a = b, then b can be substituted
for a in any expression.
If x = 7, then 2x = 2(7).
a b
= .
c c
k
k
= 8, then (2) = 8(2).
2
2
If 6 = 3t, then
6 3t
= .
3 3
When solving an algebraic equation, justify each step by using a definition,
property, or piece of given information.
2(a + 1) = −6
2a + 2 = −6
−2
−2
Given equation
Distributive Property
Subtraction Property of Equality
2a = −8
Simplify.
2a −8
=
2
2
Division Property of Equality
a = −4
Simplify.
Solve each equation. Write a justification for each step.
1.
n
− 3 = 10
6
2. 5 + x = 2x
3.
y +4
=3
7
4. 4(t − 3) = −20
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2-38
Holt McDougal Geometry
Name ________________________________________ Date __________________ Class__________________
LESSON
2-5
Reteach
Algebraic Proof continued
When writing algebraic proofs in geometry, you can also use definitions, postulates,
properties, and pieces of given information to justify the steps.
m∠JKM = m∠MKL
(5x − 12)° = 4x°
Definition of congruent angles
Substitution Property of Equality
x − 12 = 0
Subtraction Property of Equality
x = 12
Addition Property of Equality
Properties of
Congruence
Symbols
Examples
Reflexive
figure A ≅ figure A
∠CDE ≅ ∠CDE
Symmetric
If figure A ≅ figure B, then figure B ≅
figure A.
If JK ≅ LM, then LM ≅ JK .
Transitive
If figure A ≅ figure B and figure B ≅
figure C, then figure A ≅ figure C.
If ∠N ≅ ∠P and ∠P ≅ ∠Q,
then ∠N ≅ ∠Q.
Write a justification for each step.
5. CE = CD + DE
_________________________
6x = 8 + (3x + 7)
_________________________
6x = 15 + 3x
_________________________
3x = 15
_________________________
x=5
_________________________
6. m∠PQR = m∠PQS + m∠SQR
90° = 2x° + (4x − 12)°
90 = 6x − 12
_____________________________
_____________________________
_____________________________
102 = 6x
_____________________________
17 = x
_____________________________
Identify the property that justifies each statement.
7. If ∠ABC ≅ ∠DEF, then ∠DEF ≅ ∠ABC.
8. ∠1 ≅ ∠2 and ∠2 ≅ ∠3, so ∠1 ≅ ∠3.
_________________________________________
9. If FG = HJ, then HJ = FG.
________________________________________
10. WX ≅ WX
_________________________________________
________________________________________
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Holt McDougal Geometry
Addition Property of Equality
expression. So if a = b and b = c, then
a = c by the Substitution Property, and
this is also the Transitive Property.
5. Possible answer: Consider the points
A(0, 1), B(1, 0), C(0, −1), and D(−1, 0).
For reflection across the x-axis, the
image of AB is CB. AB ≅ AD , but you
cannot conclude that the image of AD is
CB for reflection across the x-axis.
Reteach
Division Property of Equality
7. Symmetric Property of Congruence
8. Transitive Property of Congruence
9. Symmetric Property of Equality
10. Reflexive Property of Congruence
Challenge
1. a. Subtr. Prop. of =
b. Mult. Prop. of =
1.
+3
+3
n
= 13
6
c. Simplify.
Add. Prop. of =
d. Distributive Property
Simplify.
n
(6) = 13(6)
6
n = 78
e. Add. Prop. of =
f. Distributive Property
Mult. Prop. of =
2. a. Add. Prop. of =
Simplify.
b. Mult. Prop. of =
2.
c. Simplify.
−x
3.
− x Subtr. Prop. of =
5=x
Simplify.
e. Subtr. Prop. of =
x=5
Sym. Prop. of =
f. Distributive Property
y +4
(7) = 3(7)
7
y + 4 = 21
−4
−4
y = 17
4.
d. Distributive Property
4t − 12 = −20
+ 12 + 12
4t = −8
4t −8
=
4 4
t = −2
3.
∠1 ≅ ∠2
m∠1 = m∠2
65° = m∠2
m∠1 + m∠2 = m∠ABC
65° + 65° = m∠ABC
130° = m∠ABC
Mult. Prop. of =
Simplify.
Subtr. Prop. of =
Simplify.
Trans. Prop. of ≅
Def. of ≅ ∠s
Subst. Prop. of =
∠ Add. Post.
Subst. Prop. of =
Simplify.
Distr. Prop.
Add Prop. of =
Simplify.
Problem Solving
1.
Div. Prop. of =
Substitution Property of Equality
1015(p + 1.39) = 3298.75
p + 1.39 = 3.25
p = $1.86
2.
C = 7.25s + 15.95a
Simplify.
298.70 = 7.25s + 15.95(6)
Subst. Prop. of =
Subtraction Property of Equality
298.70 = 7.25s + 95.7
Simplify.
Simplify.
5. Segment Addition Postulate
n(p + t) = 3298.75
Division Property of Equality
203 = 7.25s
6. Angle Addition Postulate
28 = s
Substitution Property of Equality
s = 28 students
Simplify.
3. B
Given equation
Subst. Prop. of =
Div. Prop. of =
Subtr. Prop. of =
Given equation
Subtr. Prop. of =
Div. Prop. of =
Sym. Prop. of =
4. F
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Holt McDougal Geometry