Flowchart & Paragraph Proofs
Two-column isn't the only way. Flowchart proofs show logic visually, and paragraph proofs use complete sentences — same reasoning, different presentation.
Introduction
A proof is a logical argument. The two-column format is most common in geometry class, but mathematicians often use paragraph proofs, and visual learners benefit from flowchart proofs. All three formats contain the same logic.
Past Knowledge
Two-column proofs (2.3.3). Segment & angle proofs (2.3.4-2.3.5).
Today's Goal
Write and interpret flowchart proofs and paragraph proofs.
Future Success
In higher math and college, paragraph proofs are the standard. Flowcharts enhance conceptual understanding.
Key Concepts
Flowchart Proof
- Uses boxes for statements and arrows to show logical flow.
- The reason is written below or inside each box.
- Multiple arrows can merge (parallel reasoning) or branch out.
- Read left-to-right or top-to-bottom.
Paragraph Proof
- Written in complete sentences like an essay.
- Each sentence states a fact and its reason in natural language.
- Sometimes called an “informal proof” — but it must be just as rigorous.
All Three Formats = Same Logic
A two-column, flowchart, and paragraph proof of the same theorem contain the exact same statements and reasons — just arranged differently.
Worked Examples
Vertical Angles Proof (Flowchart)
Given: and are vertical angles. Prove: .
and are vertical ∠s
Given
and are a linear pair; and are a linear pair
Definition of linear pair
and are both supp. to
Linear Pair Postulate
Congruent Supplements Thm.
Vertical Angles Proof (Paragraph)
Same theorem, paragraph format:
We are given that and are vertical angles. By definition, they are formed by two intersecting lines and share a common linear-pair partner, . Since and form a linear pair, they are supplementary by the Linear Pair Postulate. Similarly, and are supplementary. Because and are both supplementary to the same angle (), the Congruent Supplements Theorem tells us . ∎
Same Proof — Two-Column Format
Compare the same vertical angles theorem, now in two-column format:
| Statements | Reasons |
|---|---|
| and are vertical angles | Given |
| & linear pair; & linear pair | Def. of linear pair |
| supp. ; supp. | Linear Pair Postulate |
| Congruent Supplements Thm. |
All three formats — flowchart, paragraph, and two-column — contain the same 4 logical steps. Choose whichever your teacher requires or feels most natural to you.
Common Pitfalls
Paragraph Proof ≠ Essay
Don't write opinions or filler. Every sentence must contain a statement and a reason.
Flowchart Arrows in Wrong Direction
Arrows show logical flow — from known facts to the conclusion. Make sure each arrow points from cause to effect.
Real-Life Applications
Project Management Flowcharts
Flowcharts are used everywhere in business and engineering — from PERT charts in project management to decision trees in AI. The same skill of mapping logical steps visually transfers directly from geometry flowchart proofs.
Practice Quiz
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