Properties of Equality and
Congruence
Andrew Gloag
Bill Zahner
Dan Greenberg
Jim Sconyers
Lori Jordan
Victor Cifarelli
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Printed: November 21, 2012
AUTHORS
Andrew Gloag
Bill Zahner
Dan Greenberg
Jim Sconyers
Lori Jordan
Victor Cifarelli
EDITOR
Annamaria Farbizio
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C ONCEPT
Concept 1. Properties of Equality and Congruence
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Properties of Equality and
Congruence
Here you’ll review the properties of equality you learned in Algebra I, be introduced to the properties of congruence,
and learn how to use these properties.
Suppose you know that a circle measures 360 degrees and you want to find what kind of angle one-quarter of a circle
is. After completing this Concept, you’ll be able to apply the basic properties of equality and congruence to solve
geometry problems like this one.
Watch This
MEDIA
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CK-12 Propertiesof EqualityandCongruence
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James Sousa:Introduction toProof UsingProperties of Equality
Now watch this.
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James Sousa:Introduction toProof UsingProperties of Congruence
Guidance
The basic properties of equality were introduced to you in Algebra I. Here they are again:
• Reflexive Property of Equality: AB = AB
• Symmetric Property of Equality: If m6 A = m6 B, then m6 B = m6 A
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•
•
•
•
•
•
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Transitive Property of Equality: If AB = CD and CD = EF, then AB = EF
Substitution Property of Equality: If a = 9 and a − c = 5, then 9 − c = 5
Addition Property of Equality: If 2x = 6, then 2x + 5 = 6 + 5 or 2x + 5 = 11
Subtraction Property of Equality: If m6 x + 15◦ = 65◦ , then m6 x + 15◦ − 15◦ = 65◦ − 15◦ or m6 x = 50◦
Multiplication Property of Equality: If y = 8, then 5 · y = 5 · 8 or 5y = 40
18
Division Property of Equality: If 3b = 18, then 3b
3 = 3 or b = 6
Distributive Property: 5(2x − 7) = 5(2x) − 5(7) = 10x − 35
Just like the properties of equality, there are properties of congruence. These properties hold for figures and shapes.
• Reflexive Property of Congruence: AB ∼
= AB or 6 B ∼
=6 B
∼
• Symmetric Property of Congruence: If AB = CD, then CD ∼
= AB. Or, if 6 ABC ∼
= 6 DEF, then 6 DEF ∼
=
6 ABC
• Transitive Property of Congruence: If AB ∼
= CD and CD ∼
= EF, then AB ∼
= EF. Or, if 6 ABC ∼
= 6 DEF and
∼
∼
6 DEF = 6 GHI, then 6 ABC = 6 GHI
When you solve equations in algebra you use properties of equality. You might not write out the property for each
step, but you should know that there is an equality property that justifies that step. We will abbreviate “Property of
Equality” “PoE” and “Property of Congruence” “PoC” when we use these properties in proofs.
Example A
Solve 2(3x − 4) + 11 = x − 27 and write the property for each step (also called “to justify each step”).
2(3x − 4) + 11 = x − 27
6x − 8 + 11 = x − 27
Distributive Property
6x + 3 = x − 27
Combine like terms
6x + 3 − 3 = x − 27 − 3
Subtraction PoE
6x = x − 30
Simplify
6x − x = x − x − 30
Subtraction PoE
5x = −30
5x −30
=
5
5
x = −6
Simplify
Division PoE
Simplify
Example B
AB = 8, BC = 17, and AC = 20. Are points A, B, and C collinear?
Set up an equation using the Segment Addition Postulate.
AB + BC = AC
8 + 17 = 20
25 6= 20
Segment Addition Postulate
Substitution PoE
Combine like terms
Because the two sides of the equation are not equal, A, B and C are not collinear.
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Concept 1. Properties of Equality and Congruence
Example C
If m6 A + m6 B = 100◦ and m6 B = 40◦ , prove that m6 A is an acute angle.
We will use a 2-column format, with statements in one column and their reasons next to it, just like Example A.
m6 A + m6 B = 100◦
m6
m6
6
B = 40
◦
A + 40 = 100
m6
Given Information
◦
Given Information
◦
Substitution PoE
◦
A = 60
Subtraction PoE
Definition of an acute angle, m6 A < 90◦
A is an acute angle
MEDIA
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CK-12 Propertiesof EqualityandCongruence
Guided Practice
Use the given property or properties of equality to fill in the blank. x, y, and z are real numbers.
1. Symmetric: If x = 3, then ______________.
2. Distributive: If 4(3x − 8), then ______________.
3. Transitive: If y = 12 and x = y, then ______________.
Answers:
1. 3 = x
2. 12x − 32
3. x = 12
Practice
For questions 1-8, solve each equation and justify each step.
1. 3x + 11 = −16
2. 7x − 3 = 3x − 35
3. 32 g + 1 = 19
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4. 21 MN = 5
5. 5m6 ABC = 540◦
6. 10b − 2(b + 3) = 5b
7. 41 y + 56 = 31
8. 14 AB + 31 AB = 12 + 21 AB
For questions 9-11, use the given property or properties of equality to fill in the blank. x, y, and z are real numbers.
9. Symmetric: If x + y = y + z, then ______________.
10. Transitive: If AB = 5 and AB = CD, then ______________.
11. Substitution: If x = y − 7 and x = z + 4, then ______________.
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