Two-column proofs

A proof is a convincing argument that leaves no gaps. In geometry, the most common way to organize one is the two-column proof: a numbered list of statements on the left, each paired with the reason it's true on the right. You've already done this with algebra. Now the statements are about segments and angles.

The parts of a proof

Every proof starts with two things:

• Given: the facts you're allowed to assume. They're usually shown in a diagram too.
• Prove: the statement you need to reach.

Then you build a chain. Each statement must be justified by one of these:

1. Given information.
2. A definition (midpoint, angle bisector, congruent segments, right angle, and so on).
3. A property (of equality or congruence, from the last lesson).
4. A postulate: a basic fact accepted without proof.
5. A theorem that has already been proven.

The last statement is always the thing you were asked to prove.

Postulates you'll use right away

Two postulates from the foundations unit do a lot of work in proofs.

• Segment Addition Postulate: If $B$ is between $A$ and $C$, then $AB + BC = AC$.
• Angle Addition Postulate: If $P$ is in the interior of $\angle RST$, then $m\angle RSP + m\angle PST = m\angle RST$.

And these definitions, which work in both directions:

• Midpoint: $M$ is the midpoint of $\overline{AB}$ if and only if $M$ is on $\overline{AB}$ and $\overline{AM} \cong \overline{MB}$.
• Angle bisector: $\overrightarrow{BD}$ bisects $\angle ABC$ if and only if $D$ is in the interior of $\angle ABC$ and $\angle ABD \cong \angle DBC$.
• Congruent segments / angles: $\overline{AB} \cong \overline{CD}$ if and only if $AB = CD$; $\angle A \cong \angle B$ if and only if $m\angle A = m\angle B$.

A first proof

Other proof formats

A paragraph proof says the same thing in sentences: "Since $M$ is the midpoint of $\overline{AB}$, $AM = MB$. By the Segment Addition Postulate, $AM + MB = AB$, so $2 \cdot AM = AB$ and $AM = \tfrac{1}{2}AB$." A flow proof uses boxes and arrows. All three formats need the same logic; two-column proofs just make every reason visible.

Practice

Practice 1

In a two-column proof, what is the reason for the first statement almost always?

ADefinitionBGivenCReflexive PropertyDPostulate

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Practice 2

Which reason justifies: "$\overline{PQ} \cong \overline{QR}$, therefore $PQ = QR$"?

ADefinition of midpointBReflexive Property of EqualityCSegment Addition PostulateDDefinition of congruent segments

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Practice 3

$\overrightarrow{KM}$ bisects $\angle JKL$. Which statement follows directly from the definition of angle bisector?

A$\angle JKM \cong \angle MKL$B$m\angle JKL = 90^\circ$C$\angle JKM \cong \angle JKL$D$\overline{JK} \cong \overline{KL}$

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Practice 4

Which reason belongs in step 3?

#	statement	reason
1	$B$ is between $A$ and $C$	Given
2	$AB = 5$, $BC = 8$	Given
3	$AB + BC = AC$	?
4	$5 + 8 = AC$	Substitution Property of Equality
5	$AC = 13$	Simplify and Symmetric Property

ADefinition of midpointBAngle Addition PostulateCSegment Addition PostulateDTransitive Property of Equality

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Practice 5

$M$ is the midpoint of $\overline{XY}$, with $XM = 4x - 1$ and $MY = 3x + 4$. Find $XY$.

Your answer

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

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Practice 6

$\overrightarrow{QS}$ bisects $\angle PQR$. If $m\angle PQS = 2x + 12$ and $m\angle SQR = 4x - 8$, find $m\angle PQR$ in degrees.

Your answer

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

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Practice 7

Given: $m\angle 1 + m\angle 2 = 90^\circ$ and $m\angle 1 = 34^\circ$. Prove: $m\angle 2 = 56^\circ$.

#	statement	reason
1	$m\angle 1 + m\angle 2 = 90^\circ$	Given
2	$m\angle 1 = 34^\circ$	Given
3	$34^\circ + m\angle 2 = 90^\circ$	?
4	$m\angle 2 = 56^\circ$	Subtraction Property of Equality

Which reason justifies step 3?

ATransitive Property of EqualityBSubstitution Property of EqualityCAddition Property of EqualityDDefinition of complementary angles

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Practice 8

Given: $\overline{AB} \cong \overline{CD}$ and $\overline{CD} \cong \overline{EF}$. Prove: $\overline{EF} \cong \overline{AB}$. Which pair of reasons, in order, completes the proof after listing the givens?

#	statement	reason
1	$\overline{AB} \cong \overline{CD}$, $\overline{CD} \cong \overline{EF}$	Given
2	$\overline{AB} \cong \overline{EF}$	?
3	$\overline{EF} \cong \overline{AB}$	?

ASymmetric Property, then Transitive PropertyBReflexive Property, then Symmetric PropertyCTransitive Property, then Reflexive PropertyDTransitive Property, then Symmetric Property

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