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Properties of Equality and Congruence

Logical rules that allow equations to be manipulated and solved.

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Properties of Equality and Congruence

Properties of Equality and Congruence

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

For all real numbers a,b\begin{align*}a, b\end{align*}, and c\begin{align*}c\end{align*}:

Examples
Reflexive Property of Equality a=a\begin{align*}a = a\end{align*} 25=25\begin{align*}25 = 25\end{align*}
Symmetric Property of Equality a=b\begin{align*}a = b\end{align*} and b=a\begin{align*}b = a\end{align*} mP=90\begin{align*}m \angle P = 90^\circ\end{align*} or 90=mP\begin{align*}90^\circ = m \angle P\end{align*}
Transitive Property of Equality a=b\begin{align*}a = b\end{align*} and b=c\begin{align*}b = c\end{align*}, then a=c\begin{align*}a = c\end{align*} a+4=10\begin{align*}a + 4 = 10\end{align*} and 10=6+4\begin{align*}10 = 6 + 4\end{align*}, then a+4=6+4\begin{align*}a + 4 = 6 + 4\end{align*}
Substitution Property of Equality If a=b\begin{align*}a = b\end{align*}, then b\begin{align*}b\end{align*} can be used in place of a\begin{align*}a\end{align*} and vise versa. If a=9\begin{align*}a = 9\end{align*} and ac=5\begin{align*}a - c = 5\end{align*}, then 9c=5\begin{align*}9 - c = 5\end{align*}
Addition Property of Equality If a=b\begin{align*}a = b\end{align*}, then a+c=b+c\begin{align*}a + c = b + c\end{align*}. If 2x=6\begin{align*}2x = 6\end{align*}, then 2x+5=6+11\begin{align*}2x + 5 = 6 + 11\end{align*}
Subtraction Property of Equality If a=b\begin{align*}a = b\end{align*}, then ac=bc\begin{align*}a - c = b - c\end{align*}. If mx+15=65\begin{align*}m \angle x + 15^\circ = 65^\circ\end{align*}, then mx+1515=6515\begin{align*}m \angle x+15^\circ-15^\circ=65^\circ-15^\circ\end{align*}
Multiplication Property of Equality If a=b\begin{align*}a = b\end{align*}, then ac=bc\begin{align*}ac = bc\end{align*}. If y=8\begin{align*}y = 8\end{align*}, then 5y=58\begin{align*}5 \cdot y=5 \cdot 8\end{align*}
Division Property of Equality If a=b\begin{align*}a = b\end{align*}, then ac=bc\begin{align*}\frac{a}{c}=\frac{b}{c}\end{align*}. If 3b=18\begin{align*}3b=18\end{align*}, then 3b3=183\begin{align*}\frac{3b}{3}=\frac{18}{3}\end{align*}
Distributive Property a(b+c)=ab+ac\begin{align*}a(b+c)=ab+ac\end{align*} 5(2x7)=5(2x)5(7)=10x35\begin{align*}5(2x-7)=5(2x)-5(7)=10x-35\end{align*}

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

This also holds true for angles and their measures. ABCDEF\begin{align*}\angle ABC \cong \angle DEF\end{align*} if and only if mABC=mDEF\begin{align*}m \angle ABC = m \angle DEF\end{align*}. Therefore, the properties of equality apply to \begin{align*}m \angle ABC\end{align*} and \begin{align*}m \angle DEF\end{align*}.

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 \begin{align*}\overline{AB} \cong \overline{AB}\end{align*} \begin{align*}\angle ABC \cong \angle CBA\end{align*}
Symmetric Property of Congruence If \begin{align*}\overline{AB} \cong \overline{CD}\end{align*}, then \begin{align*}\overline{CD} \cong \overline{AB}\end{align*} If \begin{align*}\angle ABC \cong \angle DEF\end{align*}, then \begin{align*}\angle DEF \cong \angle ABC\end{align*}
Transitive Property of Congruence If \begin{align*}\overline{AB} \cong \overline{CD}\end{align*} and \begin{align*}\overline{CD} \cong \overline{EF}\end{align*}, then \begin{align*}\overline{AB} \cong \overline{EF}\end{align*} If \begin{align*}\angle ABC \cong \angle DEF\end{align*} and \begin{align*}\angle DEF \cong \angle GHI\end{align*}, then \begin{align*}\angle ABC \cong \angle GHI\end{align*}

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.”

Justifying the Steps of Solving an Equation

Solve \begin{align*}2(3x-4)+11=x-27\end{align*} and justify each step.

\begin{align*}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}\end{align*}

Determining Collinearity

Given points \begin{align*}A, B\end{align*}, and \begin{align*}C\end{align*}, with \begin{align*}AB = 8, BC = 17\end{align*}, and \begin{align*}AC = 20\end{align*}. Are \begin{align*}A, B\end{align*}, and \begin{align*}C\end{align*} collinear?

Set up an equation using the Segment Addition Postulate.

\begin{align*}AB + BC &= AC && \text{Segment Addition Postulate}\\ 8 + 17 &= 20 && \text{Substitution PoE}\\ 25 & \neq 20 && \text{Combine like terms}\end{align*}

Because the two sides are not equal, \begin{align*}A, B\end{align*} and \begin{align*}C\end{align*} are not collinear.

Writing a Two-Column Proof

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

Examples

Use the given property or properties of equality to fill in the blank. \begin{align*}x, y\end{align*}, and \begin{align*}z\end{align*} are real numbers.

Example 1

Symmetric: If \begin{align*}x = 3\end{align*}, then ______________.

\begin{align*} 3=x\end{align*}

Example 2

Distributive: If \begin{align*}4(3x - 8)\end{align*}, then ______________.

=\begin{align*}12x-32\end{align*}

Example 3

Transitive: If \begin{align*}y = 12\end{align*} and \begin{align*}x = y\end{align*}, then ______________.

\begin{align*}x=12\end{align*}

Review

For questions 1-8, solve each equation and justify each step.

1. \begin{align*}3x+11=-16\end{align*}
2. \begin{align*}7x-3=3x-35\end{align*}
3. \begin{align*}\frac{2}{3} g+1=19\end{align*}
4. \begin{align*}\frac{1}{2} MN=5\end{align*}
5. \begin{align*}5m \angle ABC=540^\circ\end{align*}
6. \begin{align*}10b-2(b+3)=5b\end{align*}
7. \begin{align*}\frac{1}{4}y+\frac{5}{6}=\frac{1}{3}\end{align*}
8. \begin{align*}\frac{1}{4}AB+\frac{1}{3}AB=12+\frac{1}{2}AB\end{align*}

For questions 9-12, use the given property or properties of equality to fill in the blank. \begin{align*}x, y\end{align*}, and \begin{align*}z\end{align*} are real numbers.

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

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Color Highlighted Text Notes

Vocabulary Language: English

Real Number

A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.

Right Angle Theorem

The Right Angle Theorem states that if two angles are right angles, then the angles are congruent.

Same Angle Supplements Theorem

The Same Angle Supplements Theorem states that if two angles are supplementary to the same angle then the two angles are congruent.

Vertical Angles Theorem

The Vertical Angles Theorem states that if two angles are vertical, then they are congruent.