Writing a Proof

A proof is a chain of statements. Each statement must follow from the given information or from earlier statements, and each one needs a reason. You have the pieces: definitions (like midpoint and bisector from Angles and Angle Bisectors), postulates, and the properties of equality from Properties of Equality as Reasons.

Allowed reasons: given information, definitions, postulates, properties of equality and congruence, and theorems you have already proved.
Not allowed: "it looks that way," measuring the picture, or the statement you are trying to prove.

Why it works

A drawing shows one case; a proof covers every case at once. If you measure a diagram and find two angles equal, you know about that drawing. If you prove it, you know it for every figure that fits the given facts, however it is drawn.

How to plan a proof

Work from both ends. Write the given facts at the top and the goal at the bottom. Ask: what definition or theorem would give me the goal? What would I need for that? Keep going until the two ends meet. Then write it neatly, top to bottom.

Worked example: A two-column proof

Given: $M$ is the midpoint of $\overline{AB}$. Prove: $AM = \frac{1}{2}AB$.

1. $M$ is the midpoint of $\overline{AB}$ — Given.
2. $AM = MB$ — Definition of midpoint.
3. $AM + MB = AB$ — Segment Addition Postulate.
4. $AM + AM = AB$ — Substitution Property (step 2 into step 3).
5. $2 \cdot AM = AB$ — Combine like terms.
6. $AM = \frac{1}{2}AB$ — Division Property of Equality.

Answer: Each step follows from the ones above it, so $AM = \frac{1}{2}AB$.

Two angles that fill a right angle always add to $90^\circ$. The next example proves it.

Worked example: A paragraph proof

Given: $\angle ABC$ is a right angle and $D$ is in its interior. Prove: $\angle ABD$ and $\angle DBC$ are complementary.

Because $\angle ABC$ is a right angle, $m\angle ABC = 90^\circ$ by the definition of a right angle.
$D$ is in the interior, so the Angle Addition Postulate gives $m\angle ABD + m\angle DBC = m\angle ABC$.
Substituting, $m\angle ABD + m\angle DBC = 90^\circ$.
Two angles whose measures add to $90^\circ$ are complementary, by definition.

Answer: The same reasoning in sentences: a paragraph proof is just as valid as a two-column one.

Key idea

Every statement in a proof needs a reason that comes from the given facts, a definition, a postulate, a property, or a theorem already proved. Plan from both ends, then write from the top down.

Common mistake

Using what you are trying to prove as a reason, or trusting the diagram. "The angles are equal because they look equal" and "because that is what we want to show" both prove nothing. Check each reason: could you point to the given fact, definition or theorem it comes from?

Writing a Proof

How to plan a proof

Practice

Builds on

Leads to

Sources and further reading