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2.7: Properties of Equality and Congruence

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What if you wanted to solve an equation and justify each step? What mathematical properties could you use in your justification? After completing this Concept, you be able to see how the properties of equality from Algebra I relate to geometric properties of congruence.

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CK-12 Foundation: Chapter2PropertiesofEqualityandCongruenceA

James Sousa: Introduction to Proof Using Properties of Equality

Guidance

The basic properties of equality were introduced to you in Algebra I. Here they are again:

For all real numbers a, b, and c:

Examples
Reflexive Property of Equality a = a 25 = 25
Symmetric Property of Equality a = b and b = a m \angle P = 90^\circ or 90^\circ = m \angle P
Transitive Property of Equality a = b and b = c, then a = c a + 4 = 10 and 10 = 6 + 4, then a + 4 = 6 + 4
Substitution Property of Equality If a = b, then b can be used in place of a and vise versa. If a = 9 and a - c = 5, then 9 - c = 5
Addition Property of Equality If a = b, then a + c = b + c. If 2x = 6, then 2x + 5 = 6 + 11
Subtraction Property of Equality If a = b, then a - c = b - c. If m \angle x + 15^\circ = 65^\circ, then m \angle x+15^\circ-15^\circ=65^\circ-15^\circ
Multiplication Property of Equality If a = b, then ac = bc. If y = 8, then 5 \cdot y=5 \cdot 8
Division Property of Equality If a = b, then \frac{a}{c}=\frac{b}{c}. If 3b=18, then \frac{3b}{3}=\frac{18}{3}
Distributive Property a(b+c)=ab+ac 5(2x-7)=5(2x)-5(7)=10x-35

Recall that \overline{AB} \cong \overline{CD} if and only if AB = CD. \overline{AB} and \overline{CD} represent segments, while AB and CD are lengths of those segments, which means that AB and CD are numbers. The properties of equality apply to AB and CD.

This also holds true for angles and their measures. \angle ABC \cong \angle DEF if and only if m \angle ABC = m \angle DEF. Therefore, the properties of equality apply to m \angle ABC and m \angle DEF.

Just like the properties of equality, there are properties of congruence. These properties hold for figures and shapes.

For Line Segments For Angles
Reflexive Property of Congruence \overline{AB} \cong \overline{AB} \angle ABC \cong \angle CBA
Symmetric Property of Congruence If \overline{AB} \cong \overline{CD}, then \overline{CD} \cong \overline{AB} 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} 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 logical justification for each step in your solution, 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.”

Example A

Solve 2(3x-4)+11=x-27 and 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

Given points A, B, and C, with AB = 8, BC = 17, and AC = 20. Are 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 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 \angle A is an acute angle.

We will use a two-column format, with statements in one column and their corresponding reasons in the next. This is formally called a two-column proof.

Statement Reason
1. m \angle A+m \angle B=100^\circ and m \angle B = 40^\circ Given (always the reason for using facts that are told to us in the problem)
2. m \angle A+40^\circ=100^\circ Substitution PoE
3. m \angle A = 60^\circ Subtraction PoE
4. \angle A is an acute angle Definition of an acute angle, m \angle A < 90^\circ

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter2PropertiesofEqualityandCongruenceB

Vocabulary

The properties of equality and properties of congruence are the logical rules that allow equations to be manipulated and solved.

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 9-12, 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 _________.
  4. Distributive: If 3(2x-4)=y, then_____.
  5. Given points E, F, and G and EF = 16, FG = 7 and EG = 23. Determine if E, F and G are collinear.
  6. Given points H, I and J and HI = 9, IJ = 9 and HJ = 16. Are the three points collinear? Is I the midpoint?
  7. If m \angle KLM = 56^\circ and m \angle KLM + m \angle NOP = 180^\circ, explain how \angle NOP must be an obtuse angle.

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Difficulty Level:

At Grade

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Date Created:

Jul 17, 2012

Last Modified:

Apr 17, 2014
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