Properties of Equality and Justified Steps | Geometry | LevelKite

You already know how to solve equations. In geometry, you add a new habit: you say why each step is allowed. This habit prepares you for proof, a logical argument in which every statement has a reason.

Many of those reasons are properties. A property is a rule that holds every time its conditions are met.

Start with what an equals sign means. When $a = b$ , the letters $a$ and $b$ name the same number. So if you apply the same operation to both, the results match, as long as the operation is defined. That idea gives you four properties, one for each operation.

Properties that apply an operation to both sides (a, b and c are real numbers)

Property	Rule	Example
Addition Property of Equality	If $a = b$ , then $a + c = b + c$ .	If $x - 4 = 9$ , then $x = 13$ (add $4$ to both sides).
Subtraction Property of Equality	If $a = b$ , then $a - c = b - c$ .	If $x + 4 = 9$ , then $x = 5$ (subtract $4$ from both sides).
Multiplication Property of Equality	If $a = b$ , then $ac = bc$ .	If $\frac{x}{4} = 3$ , then $x = 12$ (multiply both sides by $4$ ).
Division Property of Equality	If $a = b$ and $c \neq 0$ , then $\frac{a}{c} = \frac{b}{c}$ .	If $2x = 18$ , then $x = 9$ (divide both sides by $2$ , and $2 \neq 0$ ).

Geometry also compares figures. Two segments are congruent when they have the same length, and two angles are congruent when they have the same measure. The symbol for congruent is $\cong$ .

Watch the notation. $\overline{AB}$ is a segment, and $AB$ is its length. $\angle A$ is an angle, and $m\angle A$ is its measure in degrees. You write $\cong$ between figures and $=$ between numbers. Each definition in the table below works in both directions. You may call two figures congruent only when that is stated or proved, never because they look alike.

Figures	Congruence of figures	Equality of numbers	Reason for switching
Segments	$\overline{AB} \cong \overline{CD}$	$AB = CD$	Definition of congruent segments
Angles	$\angle A \cong \angle B$	$m\angle A = m\angle B$	Definition of congruent angles

Three properties describe how equal or congruent statements behave. Reflexive means a quantity or figure matches itself. Symmetric means you may swap the two sides. Transitive means that when two statements share a middle item, you can connect the two outer items. All three carry over from equality to congruence because congruence is defined by equal lengths or equal measures. For the exact name, add of Equality or of Congruence, as in Symmetric Property of Congruence.

The congruence rules hold for segments in the same way.

Property	Of Equality (real numbers)	Of Congruence (shown for angles)
Reflexive	$a = a$	$\angle A \cong \angle A$
Symmetric	If $a = b$ , then $b = a$ .	If $\angle A \cong \angle B$ , then $\angle B \cong \angle A$ .
Transitive	If $a = b$ and $b = c$ , then $a = c$ .	If $\angle A \cong \angle B$ and $\angle B \cong \angle C$ , then $\angle A \cong \angle C$ .

You need two more reasons.

Substitution Property of Equality: if $a = b$ , you may replace $a$ with $b$ in any expression or equation. For example, if $y = 7$ , then $2y + 1$ can be written as $2(7) + 1$ .
Distributive Property: $a(b + c) = ab + ac$ for all real numbers $a$ , $b$ and $c$ . This is not a property of equality. It is a postulate, a statement accepted as true without proof, so it counts as a reason. Read from right to left, it also justifies combining like terms: $3x + 2x = (3 + 2)x = 5x$ .

Worked example

Name the property

Name the property that justifies each statement. (a) $PQ = PQ$ . (b) If $\overline{AB} \cong \overline{CD}$ , then $\overline{CD} \cong \overline{AB}$ . (c) If $m\angle 1 = m\angle 2$ and $m\angle 2 = m\angle 3$ , then $m\angle 1 = m\angle 3$ . (d) If $AB = 6$ and $AB + BC = 10$ , then $6 + BC = 10$ . (e) If $\frac{y}{5} = 3$ , then $y = 15$ .

PQ is a length, so statement (a) is about numbers and uses an equals sign. It says a quantity equals itself. Reason: Reflexive Property of Equality.
$PQ = PQ$
Statement (b) uses the congruence symbol between segments. It swaps the two sides of a statement. Reason: Symmetric Property of Congruence.
$\overline{AB} \cong \overline{CD} \quad \text{gives} \quad \overline{CD} \cong \overline{AB}$
Statement (c) uses equals signs because angle measures are numbers. Two statements share the measure of angle 2, which connects the other two measures. Reason: Transitive Property of Equality.
$m\angle 1 = m\angle 3$
Statement (d) replaces AB with its equal value, 6, inside another equation. Reason: Substitution Property of Equality.
$AB + BC = 10 \quad \text{gives} \quad 6 + BC = 10$
Statement (e) does not show its operation, so ask what turns the first equation into the second. Multiplying both sides by 5 does, and the denominator 5 is not zero. Reason: Multiplication Property of Equality.
$5 \cdot \frac{y}{5} = 5 \cdot 3 \quad \text{so} \quad y = 15$

Answer

The reasons are (a) Reflexive Property of Equality, (b) Symmetric Property of Congruence, (c) Transitive Property of Equality, (d) Substitution Property of Equality and (e) Multiplication Property of Equality.

Now put these names to work. When you solve an equation with reasons, keep two things apart. The given is what you are told is true at the start. The solution is what you must reach, so it cannot support a step on the way there. Besides the word Given and the names above, a reason may be any definition, postulate or previously proved theorem. A theorem is a statement that has already been proved.

Key concept

Every line needs a named reason

Solving an equation is a chain of statements that starts from what is given. Each later line must follow from earlier lines. Give every line one reason: the word Given, or the exact name of a definition, postulate, property of equality or congruence, or previously proved theorem. A line cannot be its own reason, so "because $x = 4$ " never justifies the line $x = 4$ . An unstated fact or the look of a figure cannot be a reason either. Whenever you divide both sides, state that the divisor is not zero.

Worked example

Justify a two-step solution

You are given $3x + 5 = 17$ . Find $x$ and give a reason for each step.

Write the equation you are told is true. You accept it at the start, so it needs no other support. Reason: Given.
$3x + 5 = 17$
Subtract 5 from both sides to undo the added 5. Equal numbers minus the same number stay equal. Working out each side is part of this one step, so the step has one reason. Reason: Subtraction Property of Equality.
$3x + 5 - 5 = 17 - 5 \quad \text{so} \quad 3x = 12$
Divide both sides by 3 to undo the multiplication. The divisor 3 is not zero, so the division is allowed. Reason: Division Property of Equality.
$\frac{3x}{3} = \frac{12}{3} \quad \text{so} \quad x = 4$
Your steps show that if the given equation is true, then x is 4. Now check the other direction: does 4 make the equation true? Replace x with 4 in the given equation. Both sides then equal 17, so it does. Reason: Substitution Property of Equality.
$3(4) + 5 = 12 + 5 = 17$

Answer

The solution is $x = 4$ , and each step has a named reason.

Worked example

Justify a multi-step solution

You are given $9 = \frac{3(x - 1)}{2}$ . Find $x$ and give a reason for each step.

Write the equation you are told is true. Its denominator, 2, is not zero, so the fraction is defined. Reason: Given.
$9 = \frac{3(x - 1)}{2}$
Multiply both sides by 2 to clear the denominator. Reason: Multiplication Property of Equality.
$2 \cdot 9 = 2 \cdot \frac{3(x - 1)}{2} \quad \text{so} \quad 18 = 3(x - 1)$
Multiply 3 by each term inside the parentheses. Reason: Distributive Property.
$18 = 3 \cdot x - 3 \cdot 1 \quad \text{so} \quad 18 = 3x - 3$
Add 3 to both sides to isolate the term with x. Reason: Addition Property of Equality.
$18 + 3 = 3x - 3 + 3 \quad \text{so} \quad 21 = 3x$
Divide both sides by 3, which is not zero. Reason: Division Property of Equality.
$\frac{21}{3} = \frac{3x}{3} \quad \text{so} \quad 7 = x$
Swap the two sides so the variable comes first. Reason: Symmetric Property of Equality.
$x = 7$

Answer

The solution is $x = 7$ , and it checks because $\frac{3(7 - 1)}{2} = \frac{18}{2} = 9$ .

Worked example

Start from congruence statements

You are given $\overline{AB} \cong \overline{CD}$ and $\overline{CD} \cong \overline{EF}$ , with $AB = x + 11$ and $EF = 3x - 5$ . Find $x$ and give a reason for each step.

The two given statements share segment CD, so connect the outer segments. Reason: Transitive Property of Congruence.
$\overline{AB} \cong \overline{EF}$
Go from congruent segments to equal lengths. Reason: Definition of congruent segments.
$AB = EF$
Replace each length with its given expression. Reason: Substitution Property of Equality.
$x + 11 = 3x - 5$
Subtract 3x from both sides. Reason: Subtraction Property of Equality.
$x + 11 - 3x = 3x - 5 - 3x \quad \text{so} \quad -2x + 11 = -5$
Subtract 11 from both sides. Reason: Subtraction Property of Equality.
$-2x + 11 - 11 = -5 - 11 \quad \text{so} \quad -2x = -16$
Divide both sides by -2, which is not zero. Reason: Division Property of Equality.
$\frac{-2x}{-2} = \frac{-16}{-2} \quad \text{so} \quad x = 8$

Answer

The value is $x = 8$ , and it checks because $AB = 8 + 11 = 19$ and $EF = 3(8) - 5 = 19$ are equal, positive lengths.

Every step in solving an equation is a statement you can defend with a named reason. Start from what is given, name the property for the operation you actually perform, and confirm that any divisor is not zero. Next, you will use these same reasons to complete a two-column algebraic proof and to write a two-column proof for a geometric equation.