Same Math › Geometry · grades 9–10 › Flowchart and paragraph proofs
What are flowchart and paragraph proofs?
Flowchart and paragraph proofs say the same thing as two-column proofs in a different layout. A flowchart proof puts each statement in a box with its reason underneath, and arrows show which facts lead to which. A paragraph proof writes the steps as sentences: “Since AB ≅ CB is given…, the triangles are congruent by SAS.”
- Boxes: statements, with the reason written under each.
- Arrows: which facts lead to the next one.
- Paragraph: the same steps in full sentences.
Why schools teach it
Showing how the facts connect
Geometry standards ask students to prove theorems and allow any clear format (HSG.CO.C.9 and C.10). Flowcharts show which facts depend on which, something a two-column list hides.
Paragraph proofs are how mathematicians actually write. They build toward writing arguments in later math and science classes.
At the kitchen table
What to say, and what to skip
Say this
Which facts point to this box?
What's the reason under this statement?
The same reasons as a two-column proof.Can you say this step as a sentence with “since” or “because”?
Where does the arrow go next?
Skip this
- Thinking it's a different kind of math.It's the same proof, rearranged.
- Leaving reasons out of a paragraph proof.Every “since” still needs its reason.
- Drawing arrows from the conclusion backwards.Arrows point from the facts toward what they prove.
See it in action
Turning a two-column proof into a paragraph
Start with the given
We are given that AB ≅ CB and BD bisects ∠ABC.Each step becomes a clause with its reason
So ∠ABD ≅ ∠CBD by the definition of angle bisector, and BD ≅ BD by the Reflexive Property.End with the conclusion and its reason
So △ABD ≅ △CBD by SAS.
Words like “since,” “so” and “therefore” do the work of the arrows.
Free practice sheet
Make a flowchart or paragraph proof sheet
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Parents also ask
Which format do teachers prefer?
It depends on the class and the textbook. Many teach all three and let students choose on tests.
Is a flowchart proof easier?
Many kids find it easier because they can see which facts lead where.
Do the reasons change between formats?
No. The statements and reasons are the same; only the layout changes.
What makes a good paragraph proof?
Every claim has its reason, in order, ending with what you set out to prove.