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Three Types of Geometric Proofs You Need to Know

Geometric proofs are given statements that prove a mathematical concept is true. In order for a proof to be proven true, it has to include multiple steps. These steps are made up of reasons and statements.

There are many types of geometric proofs, including two-column proofs, paragraph proofs, and flowchart proofs. We’ll walk you through each type.

Two-Column Proof

geometric proofs: Diagram showing a two column proof Credit: Powered by Desmos

The columns above show how the shared midpoint, vertical angles of triangles FGH and IJH, and SAS (Side Angle Side) theorem prove the triangles are congruent.

In two-column proofs, the first column has a chronological list of steps. The second column uses deductive reasoning to create a complementary justification for each step. These justifications are either definitions, postulates (assumptions based on mathematical reasoning), or theorems (rules demonstrated through formulas).

Since two-column proofs have a clear-cut way of displaying every step, they're commonly used in high school geometry classes.

Paragraph Proof

geometric proofs: Diagram showing a paragraph proof Credit: Powered by Desmos

The paragraph above explains that because of the congruence of angles FHG and IHJ and because line segments FI and GJ have a shared midpoint of H, FGH and DEC are congruent triangles.

Paragraph proofs are comprehensive paragraphs that explain the process of each proof. Like two-column proofs, they have multiple steps and justifications. But instead of columns, the given information is formatted like a word problem — written out in long-hand format.

Paragraph proofs need to be written in chronological order, showing that each step allows the next statement to be true. Each step needs to be supported by a definition, theorem, or postulate. Since paragraph proofs are wordier and harder to follow, they're more commonly used by college educators.

Flowchart Proof

Diagram showing a flowchart proof Credit: Powered by Desmos

The chart above uses arrows and boxes to prove that FGH and IJH have congruent angles, congruent sides, and are ultimately congruent triangles.

Flowchart proofs demonstrate geometry proofs by using boxes and arrows. In this method, statements are written inside boxes and reasons are written beneath each box.

Unlike the other two proofs, flowcharts don't require you to write out every step and justification. Instead, boxes and arrows provide a detailed view of each proof, making it easier to understand how each statement leads to a logical argument.

Organizing Your Geometric Proofs

Two-column, paragraph, and flowchart proofs are three of the most common geometric proofs. They each offer different ways of organizing reasons and statements so that each proof can be easily explained.

For more useful geometric concepts, check out our articles on the pythagorean theorem, quadratic regression, and one-to-one-functions.

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