Math Core

When you solved equations in algebra, you added, subtracted and divided without thinking too hard about why each move was allowed. In geometry, every step of an argument needs a reason. The properties in this lesson are the reasons you'll cite most often, both for equations and for congruent figures.

Properties of equality

These properties hold for any real numbers $a$ , $b$ and $c$ . Most of them say the same thing: do the same thing to both sides and the equation stays true.

property	statement
Addition Property	If $a = b$ , then $a + c = b + c$ .
Subtraction Property	If $a = b$ , then $a - c = b - c$ .
Multiplication Property	If $a = b$ , then $ac = bc$ .
Division Property	If $a = b$ and $c \ne 0$ , then $\dfrac{a}{c} = \dfrac{b}{c}$ .
Substitution Property	If $a = b$ , then $b$ can replace $a$ in any expression or equation.
Distributive Property	$a(b + c) = ab + ac$

Reflexive, symmetric and transitive

Three more properties describe how equality itself behaves.

Reflexive: $a = a$ . Everything equals itself.
Symmetric: If $a = b$ , then $b = a$ . You can flip an equation.
Transitive: If $a = b$ and $b = c$ , then $a = c$ . Equal to the same thing means equal to each other.

These might seem too obvious to write down, but a proof has to state every step. The reflexive property, for example, is exactly what you'll cite later when two triangles share a side.

Properties of congruence

Recall that segments are congruent when they have equal lengths, and angles are congruent when they have equal measures. So $\overline{AB} \cong \overline{CD}$ means $AB = CD$ (a statement about numbers), and $\angle P \cong \angle Q$ means $m\angle P = m\angle Q$ .

Because congruence is defined through equality, it inherits the reflexive, symmetric and transitive properties.

Properties of congruence

For segments and angles:

property	segments	angles
Reflexive	$\overline{AB} \cong \overline{AB}$	$\angle A \cong \angle A$
Symmetric	If $\overline{AB} \cong \overline{CD}$ , then $\overline{CD} \cong \overline{AB}$ .	If $\angle A \cong \angle B$ , then $\angle B \cong \angle A$ .
Transitive	If $\overline{AB} \cong \overline{CD}$ and $\overline{CD} \cong \overline{EF}$ , then $\overline{AB} \cong \overline{EF}$ .	If $\angle A \cong \angle B$ and $\angle B \cong \angle C$ , then $\angle A \cong \angle C$ .

Common mistake

Use $=$ between numbers and $\cong$ between figures. Write $AB = CD$ (lengths) or $\overline{AB} \cong \overline{CD}$ (segments), but never $\overline{AB} = \overline{CD}$ or $AB \cong CD$ . Likewise, write $m\angle 1 = m\angle 2$ or $\angle 1 \cong \angle 2$ , not $\angle 1 = \angle 2$ . Also: "Addition Property" and "Substitution" are properties of equality only. You add numbers, not segments.

Writing an algebraic proof

An algebraic proof is solving an equation while naming the property that justifies each step. It's great practice for geometry proofs, which use the same format.

Worked example: Solving with reasons

Solve $4x - 7 = 21$ and justify each step.

statement	reason
$4x - 7 = 21$	Given
$4x = 28$	Addition Property of Equality
$x = 7$	Division Property of Equality

Each step does one thing: first add $7$ to both sides, then divide both sides by $4$ .

Worked example: A longer algebraic proof

Solve $3(x + 2) = 5x - 8$ with reasons.

statement	reason
$3(x + 2) = 5x - 8$	Given
$3x + 6 = 5x - 8$	Distributive Property
$6 = 2x - 8$	Subtraction Property of Equality
$14 = 2x$	Addition Property of Equality
$7 = x$	Division Property of Equality
$x = 7$	Symmetric Property of Equality

The last line just flips the equation so $x$ is on the left. That flip has a name, so it gets cited.

Worked example: Using geometry facts

Point $B$ lies between $A$ and $C$ , with $AB = 2x + 1$ , $BC = x + 5$ and $AC = 24$ . Find $x$ with reasons.

statement	reason
$AB + BC = AC$	Segment Addition Postulate
$(2x + 1) + (x + 5) = 24$	Substitution Property of Equality
$3x + 6 = 24$	Simplify (combine like terms)
$3x = 18$	Subtraction Property of Equality
$x = 6$	Division Property of Equality

Check: $AB = 13$ and $BC = 11$ , and $13 + 11 = 24$ .

Worked example: Naming the property

Name the property that justifies each statement.

If $m\angle 1 = m\angle 2$ and $m\angle 2 = 40^\circ$ , then $m\angle 1 = 40^\circ$ .
If $\angle R \cong \angle S$ , then $\angle S \cong \angle R$ .
$\overline{XY} \cong \overline{XY}$

Solutions.

Transitive Property of Equality (you could also call it Substitution: $40^\circ$ replaces $m\angle 2$ ).
Symmetric Property of Congruence.
Reflexive Property of Congruence.

Tip

To tell symmetric from transitive, count the statements. Symmetric uses one fact and flips it. Transitive uses two facts that share a middle term.

Practice

Practice 1

Which property justifies: If $x - 9 = 4$ , then $x = 13$ ?

Practice 2

Which property justifies: If $\overline{PQ} \cong \overline{RS}$ and $\overline{RS} \cong \overline{TU}$ , then $\overline{PQ} \cong \overline{TU}$ ?

Practice 3

Which property justifies $\angle DEF \cong \angle DEF$ ?

Practice 4

Which statement is written correctly?

Practice 5

Solve $5(x - 3) = 30$ . (A full proof would give reasons: Given, Distributive Property, Addition Property, Division Property.)

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

In this algebraic proof, which reason belongs in Step 3?

step	statement	reason
1	$\dfrac{x}{3} + 2 = 8$	Given
2	$\dfrac{x}{3} = 6$	Subtraction Property of Equality
3	$x = 18$	?

Practice 7

$\angle ABD$ and $\angle DBC$ are adjacent, with $m\angle ABD = 3x - 5$ , $m\angle DBC = 2x + 10$ and $m\angle ABC = 130^\circ$ . By the Angle Addition Postulate, $m\angle ABD + m\angle DBC = m\angle ABC$ . Find $x$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

Given $AB = CD$ , $AB = 4x - 3$ and $CD = 2x + 19$ . Find $x$ . Which property lets you write $4x - 3 = 2x + 19$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.