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# Properties of Equality and Congruence

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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.

Now watch this.

### 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.

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

### 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 ______________.

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 ______________.