# 2.14: Properties of Equality and Congruence

Logical rules involving equality and congruence that allow equations to be manipulated and solved.

## Properties of Equality and Congruence

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 $$\dfrac{3b}{3}=\dfrac{18}{3}$$ 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.

Suppose you know that a circle measures 360 degrees and you want to find what kind of angle one-quarter of a circle is.

For Examples 1 and 2, use the given property of equality to fill in the blank. $$x$$ and $$y$$ are real numbers.

Example $$\PageIndex{1}$$

Distributive: If $$4(3x−8)$$, then ______________.

Solution

$$12x−32$$

Example $$\PageIndex{2}$$

Transitive: If $$y=12$$ and $$x=y$$, then ______________

Solution

$$x=12$$

Example $$\PageIndex{3}$$

Solve $$2(3x−4)+11=x−27$$ and write the property for each step (also called “to justify each step”).

Solution

\begin{align*} 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 & Simplify\\ \dfrac{5x}{5} &= \dfrac{−30}{5}& Division \: PoE\\ x &= −6 &Simplify \end{align*}

Example $$\PageIndex{4}$$

$$AB=8$$, $$BC=17$$, and $$AC=20$$. Are points $$A$$, $$B$$, and $$C$$ collinear?

Solution

Set up an equation using the Segment Addition Postulate.

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

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

Example $$\PageIndex{5}$$

If $$m\angle A+m\angle B=100^{\circ}$$ and $$m\angle B=40^{\circ}$$, prove that $$m\angle A$$ is an acute angle.

Solution

We will use a 2-column format, with statements in one column and their reasons next to it, just like Example A.

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

## Review

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

1. $$3x+11=−16$$
2. $$7x−3=3x−35$$
3. $$\dfrac{2}{3}g+1=19$$
4. $$\dfrac{1}{2}MN=5$$
5. $$5m\angle ABC=540^{\circ}$$
6. $$10b−2(b+3)=5b$$
7. $$\dfrac{1}{4}y+\dfrac{5}{6}=\dfrac{1}{3}$$
8. $$\dfrac{1}{4}AB+\dfrac{1}{3}AB=12+\dfrac{1}{2}AB$$

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 ______________.

## Vocabulary

Term Definition
properties of equality Together with properties of congruence, the logical rules that allow equations to be manipulated and solved.
Addition Property of Inequality 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 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 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 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 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 $$\overline{AB}\cong \overline{AB}$$ or $$\angle B\cong \angle B$$
Reflexive Property of Equality Any algebraic or geometric item is equal in value to itself.
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.
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 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 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 $$\angle ABC\cong \angle GHI$$
Transitive Property of Equality If $$a = 5$$, and $$b = 5$$, then $$a = b$$.
Vertical Angles Theorem The Vertical Angles Theorem states that if two angles are vertical, then they are congruent.