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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

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

CK-12 Properties of Equality and Congruence

James Sousa: Introduction to Proof Using Properties of Equality

Now watch this.

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\begin{align*}AB = AB\end{align*}
• Symmetric Property of Equality: If mA=mB\begin{align*}m\angle A = m \angle B\end{align*}, then mB=mA\begin{align*}m \angle B = m \angle A\end{align*}
• Transitive Property of Equality: If AB=CD\begin{align*}AB = CD\end{align*} and CD=EF\begin{align*}CD = EF\end{align*}, then AB=EF\begin{align*}AB = EF\end{align*}
• Substitution Property of Equality: 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 2x=6\begin{align*}2x = 6\end{align*}, then 2x+5=6+5\begin{align*}2x + 5 = 6 + 5\end{align*} or 2x+5=11\begin{align*}2x + 5 = 11\end{align*}
• Subtraction Property of Equality: 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*} or mx=50\begin{align*}m\angle x = 50^\circ\end{align*}
• Multiplication Property of Equality: If y=8\begin{align*}y = 8\end{align*}, then 5y=58\begin{align*}5 \cdot y = 5 \cdot 8\end{align*} or 5y=40\begin{align*}5y = 40\end{align*}
• Division Property of Equality: If 3b=18\begin{align*}3b=18\end{align*}, then 3b3=183\begin{align*}\frac{3b}{3}=\frac{18}{3}\end{align*} or b=6\begin{align*}b = 6\end{align*}
• Distributive Property: 5(2x7)=5(2x)5(7)=10x35\begin{align*}5(2x-7)=5(2x)-5(7)=10x-35\end{align*}

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

• Reflexive Property of Congruence: AB¯¯¯¯¯AB¯¯¯¯¯\begin{align*}\overline{AB} \cong \overline{AB}\end{align*} or BB\begin{align*}\angle B \cong \angle B\end{align*}
• Symmetric Property of Congruence: If AB¯¯¯¯¯CD¯¯¯¯¯\begin{align*}\overline{AB} \cong \overline{CD}\end{align*}, then CD¯¯¯¯¯AB¯¯¯¯¯\begin{align*}\overline{CD} \cong \overline{AB}\end{align*}. Or, if ABCDEF\begin{align*}\angle ABC \cong \angle DEF\end{align*}, then DEFABC\begin{align*}\angle DEF \cong \angle ABC\end{align*}
• Transitive Property of Congruence: If AB¯¯¯¯¯CD¯¯¯¯¯\begin{align*}\overline{AB} \cong \overline{CD}\end{align*} and CD¯¯¯¯¯EF¯¯¯¯¯\begin{align*}\overline{CD} \cong \overline{EF}\end{align*}, then AB¯¯¯¯¯EF¯¯¯¯¯\begin{align*}\overline{AB} \cong \overline{EF}\end{align*}. Or, if ABCDEF\begin{align*}\angle ABC \cong \angle DEF\end{align*} and DEFGHI\begin{align*}\angle DEF \cong \angle GHI\end{align*}, then ABCGHI\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 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\begin{align*}PoE\end{align*}” and “Property of Congruence” “PoC\begin{align*}PoC\end{align*}” when we use these properties in proofs.

#### Example A

Solve 2(3x4)+11=x27\begin{align*}2(3x-4)+11=x-27\end{align*} and write the property for each step (also called “to justify each step”).

#### Example B

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

Set up an equation using the Segment Addition Postulate.

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

#### Example C

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*}m \angle A\end{align*} 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.

CK-12 Properties of Equality and Congruence

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### Guided Practice

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 = 3\end{align*}, then ______________.

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

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

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

2. \begin{align*}12x-32\end{align*}

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

### Explore More

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 9-11, 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 ______________.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 2.6.

### Vocabulary Language: English Spanish

properties of equality

properties of equality

Together with properties of congruence, the logical rules that allow equations to be manipulated and solved.

You can add a quantity to both sides of an inequality and it does not change the sense of the inequality. If $x > 3$, then $x+2 > 3+2$.
distributive property

distributive property

The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, $a(b + c) = ab + ac$.
Division Property of Inequality

Division Property of Inequality

The division property of inequality states that two unequal values divided by a positive number retain the same relationship. Two unequal values divided by a negative number result in a reversal of the relationship.
Multiplication Property of Equality

Multiplication Property of Equality

The multiplication property of equality states that if the same constant is multiplied to both sides of the equation, the equality holds true.
Real Number

Real Number

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

Reflexive Property of Congruence

$\overline{AB} \cong \overline{AB}$ or $\angle B \cong \angle B$
Reflexive Property of Equality

Reflexive Property of Equality

Any algebraic or geometric item is equal in value to itself.
Right Angle Theorem

Right Angle Theorem

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

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.
Substitution Property of Equality

Substitution Property of Equality

If a variable is equal to a specified amount, that amount can be directly substituted into an equation for the given variable.
Subtraction Property of Equality

Subtraction Property of Equality

The subtraction property of equality states that you can subtract the same quantity from both sides of an equation and it will still balance.
Symmetric Property of Congruence

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

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 $\angle ABC \cong \angle GHI$
Transitive Property of Equality

Transitive Property of Equality

If a = 5, and b = 5, then a = b.
Vertical Angles Theorem

Vertical Angles Theorem

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