In Why Each Step Keeps the Balance you saw that each step of solving an equation keeps the balance. A geometry proof asks you to name the rule behind each step. The rules are the properties of equality, and they work the same way for lengths and angle measures as for $x$. They are reliable for the reasons you met in If-Then Statements: each one is an if-then statement that is always true.
| Property | If… | then… |
|---|---|---|
| Addition | $a = b$ | $a + c = b + c$ |
| Subtraction | $a = b$ | $a - c = b - c$ |
| Multiplication | $a = b$ | $ac = bc$ |
| Division | $a = b$ and $c \ne 0$ | $\frac{a}{c} = \frac{b}{c}$ |
| Substitution | $a = b$ | $b$ may replace $a$ in any expression |
| Distributive | (always) | $a(b + c) = ab + ac$ |
| Reflexive | (always) | $a = a$ |
| Symmetric | $a = b$ | $b = a$ |
| Transitive | $a = b$ and $b = c$ | $a = c$ |
Por qué funciona
Each property says: if two quantities are equal, then doing the same thing to both keeps them equal. A balance that is level stays level when you add the same weight to both pans. That is why a solution found this way is guaranteed, not just likely.
Idea clave
Write each step of a solution with the property that justifies it. Undoing addition uses the Subtraction Property, undoing multiplication uses the Division Property, and replacing a quantity with an equal one is Substitution.
Ejemplo resuelto: Solving with reasons
Solve $3(x - 4) = 2x + 1$ and justify each step.
- $3(x - 4) = 2x + 1$ — Given.
- $3x - 12 = 2x + 1$ — Distributive Property.
- $x - 12 = 1$ — Subtraction Property of Equality (subtract $2x$).
- $x = 13$ — Addition Property of Equality (add 12).
Respuesta: $x = 13$
Ejemplo resuelto: Angles instead of $x$
$\angle 1$ and $\angle 2$ are complementary, and $m\angle 1 = 35^\circ$. Justify finding $m\angle 2$.
- $m\angle 1 + m\angle 2 = 90^\circ$ — Definition of complementary angles.
- $35 + m\angle 2 = 90$ — Substitution Property.
- $m\angle 2 = 55^\circ$ — Subtraction Property of Equality.
Respuesta: $m\angle 2 = 55^\circ$
Error común
Mixing up the Symmetric and Transitive Properties. Symmetric only swaps the two sides of one equation ($AB = CD$, so $CD = AB$). Transitive links two equations through a shared middle ($a = b$ and $b = c$, so $a = c$). Also, "combine like terms" simplifies one side; it is not a property of equality, since nothing is done to both sides.