# Apply Bubble Logic to Two-Column Proofs

You can add bubbles and arrows to a proof solution to show the connections between the statements and the reasons. Although you won’t be asked to do this when you solve a proof, it’s a great way to help you understand how proofs work.

Here’s an example of a bubble proof.

Note how the bubbles and arrows show how the logic flows through the proof.

In a two-column proof,

- The idea in the
*if*clause of each reason must come from the statement column somewhere*above*the reason. - The idea in the
*then*clause of each reason must match the idea in the statement on the*same line*as the reason.

The arrows and bubbles in the figure show how this incredibly important logical structure works.

Here’s another example to show you how a deductive argument all hangs together. The following proof establishes that Clyde the Clydesdale won’t be giving an address at your high school commencement. Here’s the basic argument:

**Clyde is a Clydesdale.****Therefore, Clyde is a horse.**(Because all Clydesdales are horses.)**Therefore, Clyde can’t talk.**(Because horses can’t talk.)**Therefore, Clyde can’t give a commencement address.**(Because something that doesn’t talk can’t give a commencement address.)**Therefore, Clyde won’t be giving an address at your high school commencement.**(Because something that can’t give a commencement address won’t be giving one at your high school commencement.)

Now take a look at what this argument or proof would look like in the standard two-column geometry proof format with the reasons written in *if-then* form. When reasons are written this way, you can see how the chain of logic flows.

*Given:* Clyde is a Clydesdale.

*Prove:* Clyde won’t be giving an address at your high school commencement.

Follow the arrows from bubble to bubble. Note again that the idea in the *if* clause of each reason connects to the same idea in the statement column *above* the line of the reason; the idea in the *then* clause of each reason connects to the same idea in the statement column on the *same line* as the reason.