Paragraph Proofs

Learn to write a proof as connected sentences with signal words like since and therefore, convert two-column proofs, and check a proof for gaps.

Advanced30 min18 problems
Practice Problems

Definition

A paragraph proof is a proof written as connected sentences instead of a table. It contains exactly the same logic as a two-column proof: every claim (statement) still needs a reason, but the reason is written inside the sentence.
A good paragraph proof has three parts:
  1. 1.Start: state what is given and name every object with its own letter (*Let aa and bb be odd integers.*)
  2. 2.Chain: each sentence makes one claim and gives its reason (a definition, a property of equality, an earlier theorem).
  3. 3.End: the last sentence states exactly what had to be proved, often followed by QED or ■\blacksquare.
Signal words glue the sentences together:
  • *since*, *because*, *by the definition of* introduce a reason
  • *so*, *therefore*, *thus*, *it follows that* introduce a new claim
given  ⇒  claim + reason  ⇒  …  ⇒  what was to be proved\text{given} \;\Rightarrow\; \text{claim + reason} \;\Rightarrow\; \dots \;\Rightarrow\; \text{what was to be proved}

Try it now

∠1\angle 1 and ∠2\angle 2 form a linear pair, and ∠1=72∘\angle 1 = 72^\circ. How many degrees is ∠2\angle 2?

Worked Examples

A two-column proof shows: given 3x+5=203x + 5 = 20, prove x=5x = 5. Its rows are (1) 3x+5=203x + 5 = 20, given; (2) 3x=153x = 15, subtraction property of equality; (3) x=5x = 5, division property of equality. Rewrite it as a paragraph proof.
1

Turn the first row into an opening sentence

Row 1 is the given fact, so the paragraph starts: *We are given that 3x+5=203x + 5 = 20.* → Opening sentence written
2

Join row 2 with its reason

*Subtracting 5 from both sides gives 3x=153x = 15 by the subtraction property of equality.* → 3x=153x = 15
3

Join row 3 with its reason

*Dividing both sides by 3 gives x=5x = 5 by the division property of equality.* → x=5x = 5
4

State the conclusion

*Therefore x=5x = 5.* The word therefore marks the conclusion, the last row of the table. → Paragraph proof complete

Common Mistakes

Writing a chain of equations with no words

Why it's wrong: A list like a=2m+1a = 2m + 1, a+b=2(m+n+1)a + b = 2(m + n + 1) does not say where each line comes from or what it proves. The reader has to guess the reasons.

Correct: Attach a reason to every claim (*Since aa is odd, a=2m+1a = 2m + 1...*) and finish with a sentence that states the conclusion.

Using the same letter for two different numbers

Why it's wrong: Writing a=2k+1a = 2k + 1 and b=2k+1b = 2k + 1 forces a=ba = b, so the proof only covers sums like 7+77 + 7.

Correct: Give every object its own variable: a=2m+1a = 2m + 1 and b=2n+1b = 2n + 1.

Proving with examples

Why it's wrong: 3+5=83 + 5 = 8 and 7+9=167 + 9 = 16 are even, but a few examples can never cover all odd numbers.

Correct: Use variables that stand for every possible case. Examples are useful only as a check after the proof.

Using the conclusion as a reason

Why it's wrong: Writing *Since a+ba + b is even, it equals 2(m+n+1)2(m + n + 1)* assumes exactly what you are trying to prove (circular reasoning).

Correct: The words since and because may only point to facts you already have: the given, definitions, properties or earlier sentences.

Interactive Visual

Angle Explorer

1234
∠1 = 65°∠2 = 115°∠3 = 65°∠4 = 115°

∠1 = ∠3 = 65°, ∠2 = ∠4 = 115°

Vertical angles are opposite each other where two lines cross. They are always equal.

Drag the blue point to turn the second line and watch the four angles.

Interactive Sandbox

Expression Calculator

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Practice Problems

18 problems
Problem 1 of 18
Easy
In a paragraph proof, which signal word introduces a new claim (a conclusion) rather than a reason?

Why It Matters

Most proofs you will read in textbooks, competitions and university courses are paragraph proofs, so learning to write and read them is essential:
  • Exams and competitions: graders award points for the reasoning, and a clear paragraph shows every step
  • Communication: a paragraph proof reads like an explanation you could give a classmate
  • Checking work: writing the reason next to each claim exposes gaps that a bare list of equations hides
  • Beyond math: lawyers, scientists and programmers build arguments the same way, claim by claim with a reason for each
Once you can move freely between the two-column format and a paragraph, you can choose whichever fits the problem.

Real World Applications

Calendar Puzzles

Any three days next to each other in a calendar row are consecutive integers, so the paragraph proof from this lesson explains a classic number trick.

Example:

Pick 12, 13 and 14: the sum is 39=3×1339 = 3 \times 13, three times the middle day.

1Try It Yourself

A friend picks three consecutive dates, 23, 24 and 25, and you announce the sum without adding them one by one.

What is the sum of the three dates?

Step 1: Write the mathematical expression

Use the proof: the sum is 3 times the middle number.

Scissors and Crossing Roads

Where two straight objects cross, like scissor blades, crossing roads or the beams of a roof truss, the vertical angles theorem lets builders measure one angle and know the opposite one.

Example:

If the blades of open scissors make a 35∘35^\circ angle, the handles make a 35∘35^\circ angle too.

2Try It Yourself

Two straight roads cross. The angle between them on the north side is 62∘62^\circ.

What are the measures of the angle next to it and the angle opposite it?

Step 1: Write the mathematical expression

Use one linear pair, then the next one:

Packing in Pairs

Shops and warehouses often pack items two per box. The proof that an even number times any integer is even tells you when a grid of items splits into pairs with nothing left over.

Example:

A tray with 8 rows of 7 cookies holds 56=2×2856 = 2 \times 28 cookies, so they fill 28 bags of two.

3Try It Yourself

A floor is covered with 6 rows of 13 tiles. The tiles were delivered in boxes of 2.

How many full boxes did the floor use, and how many tiles are left over?

Step 1: Write the mathematical expression

Write the total as 2(kb)2(kb) with 6=2×36 = 2 \times 3:

Key Takeaways

  • 1A paragraph proof states the same claims and reasons as a two-column proof, written as connected sentences
  • 2Start with the given facts and give every object its own letter, for example a=2m+1a = 2m + 1 and b=2n+1b = 2n + 1
  • 3Since, because and by definition introduce reasons; so, therefore and it follows that introduce new claims
  • 4Even numbers are 2k2k, odd numbers are 2k+12k + 1, and three consecutive integers add up to 3(n+1)3(n + 1)
  • 5Vertical angles are equal because each one forms a linear pair (180∘180^\circ) with the same angle between them
  • 6To check a proof, test every sentence: is the claim justified, is anything assumed that was not given, does the last sentence state the goal?

Frequently Asked Questions

No. Both formats contain the same logic. A paragraph proof is only a different layout: every claim still needs a reason, it is simply written inside the sentence instead of in a second column.
No. Both formats contain the same logic. A paragraph proof is only a different layout: every claim still needs a reason, it is simply written inside the sentence instead of in a second column.
Long enough that a classmate could follow every step without asking why. For the proofs in this lesson, 3 to 6 sentences is typical. Simple algebra, like factoring out 2, does not need its own sentence.
Only as a check, clearly separated from the proof. A sentence like *for example, 3+5=83 + 5 = 8* shows the idea, but the proof itself must use variables that cover every case.
Use since or because before a reason, and so, therefore or it follows that before a new claim. Vary them so the paragraph reads naturally, but make sure every claim has a reason somewhere in its sentence.

Glossary

Paragraph proof
A proof written as connected sentences, each claim followed or preceded by its reason
Two-column proof
A proof written as a table with statements on the left and reasons on the right
Signal word
A word such as since, because, so or therefore that shows how a sentence connects to the previous ones
Linear pair
Two adjacent angles that together form a straight line; their measures add up to 180∘180^\circ
Vertical angles
Two opposite angles formed by two intersecting lines; they are always equal
Consecutive integers
Integers that follow each other without gaps, such as nn, n+1n + 1, n+2n + 2

Formula Card

Even integer

a=2ka = 2k

for some integer $k$

Odd integer

a=2k+1a = 2k + 1

for some integer $k$

Three consecutive integers

n+(n+1)+(n+2)=3(n+1)n + (n + 1) + (n + 2) = 3(n + 1)

The sum is always 3 times the middle number

Linear pair and vertical angles

∠1+∠2=180∘,∠1=∠3\angle 1 + \angle 2 = 180^\circ, \quad \angle 1 = \angle 3

Angles of a linear pair add up to $180^\circ$; vertical angles are equal

Today's challengeOne problem, three tries

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Introduction to Mathematical Proofs

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Direct Proof

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Two-Column Proofs

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Geometric Proofs

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