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Properties of Equality and Congruence

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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.

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CK-12 Properties of Equality and Congruence

James Sousa: Introduction to Proof Using Properties of Equality

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James Sousa: Introduction to Proof Using Properties 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 m\angle A = m \angle B , then m \angle B = m \angle A
  • 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 m \angle x + 15^\circ = 65^\circ , then m\angle x+15^\circ - 15^\circ = 65^\circ - 15^\circ or m\angle x = 50^\circ
  • Multiplication Property of Equality : If y = 8 , then 5 \cdot y = 5 \cdot 8 or 5y = 40
  • Division Property of Equality : If 3b=18 , then \frac{3b}{3}=\frac{18}{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 : \overline{AB} \cong \overline{AB} or \angle B \cong \angle B
  • Symmetric Property of Congruence : If \overline{AB} \cong \overline{CD} , then \overline{CD} \cong \overline{AB} . Or, if \angle ABC \cong \angle DEF , then \angle DEF \cong \angle ABC
  • Transitive Property of Congruence : If \overline{AB} \cong \overline{CD} and \overline{CD} \cong \overline{EF} , then \overline{AB} \cong \overline{EF} . Or, if \angle ABC \cong \angle DEF and \angle DEF \cong \angle GHI , then \angle ABC \cong \angle 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 && \text{Distributive Property}\\6x+3 & = x-27 && \text{Combine like terms}\\6x+3-3& =x-27-3 && \text{Subtraction} \ PoE\\6x& =x-30 && \text{Simplify}\\6x-x& =x-x-30 && \text{Subtraction} \ PoE\\5x& =-30 && \text{Simplify}\\\frac{5x}{5}& =\frac{-30}{5} && \text{Division} \ PoE\\x& =-6 && \text{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 && \text{Segment Addition Postulate}\\8 + 17 & = 20 && \text{Substitution} \ PoE\\25 & \neq 20 && \text{Combine like terms}

Because the two sides of the equation are not equal, A, B and C are not collinear.

Example C

If m \angle A + m \angle B = 100^\circ and m \angle B = 40^\circ , prove that m \angle 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.

m \angle A + m\angle B & = 100^\circ && \text{Given Information}\\m \angle B & = 40^\circ && \text{Given Information}\\m \angle A + 40^\circ & = 100^\circ && \text{Substitution} \ PoE\\m \angle A & = 60^\circ && \text{Subtraction} \ PoE\\\angle A \ \text{is an acute} & \ \text{angle} && \text{Definition of an acute angle}, m\angle A < 90^\circ

CK-12 Properties of Equality and Congruence

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. \frac{2}{3}g+1=19
  4. \frac{1}{2} MN = 5
  5. 5m \angle ABC = 540^\circ
  6. 10b-2(b+3)=5b
  7. \frac{1}{4}y+\frac{5}{6}=\frac{1}{3}
  8. \frac{1}{4}AB+\frac{1}{3}AB=12+\frac{1}{2}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.

  1. Symmetric: If x + y = y + z , then ______________.
  2. Transitive: If AB = 5 and AB = CD , then ______________.
  3. Substitution: If x = y - 7 and x = z + 4 , then ______________.

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