Method

Paragraph Proof

A paragraph proof, or informal proof, is a way of presenting a mathematical proof that consists of statements and reasons written as complete sentences in a paragraph. The reasons can be postulates, theorems, or other mathematical reasoning that the reader is assumed to be able to follow without difficulty. For example, consider the following prompt.

Let C be a point on AE, B be a point on AC, and D be a point on CE, such that AB ≅ DE and BC ≅ CD. Prove that AC ≅ CE.

The following steps can be used to prove this particular statement.

Draw a Diagram

According to the given information, C is a point on AE.

Also, it is given that B is a point on AC and D is a point on CE.

From the last piece of given information, AB is congruent to DE and BC is congruent to CD.

Apply the Definition of Congruence

The given congruence statements imply that the congruent segments have equal lengths. AB ≅ DE and BC ≅ CD ⇓ AB= DE and BC= CD

Use the Segment Addition Postulate

The Segment Addition Postulate says that the length of a segment is the sum of the lengths of its parts. AC&= AB+ BC CE&= CD+ DE These relationships can be visualized on the diagram.

Use the Commutative Property of Addition

The Commutative Property of Addition guarantees that the order of the terms in a sum can be changed. CE = CD + DE ⇓ CE = DE + CD

Use the Substitution Property of Equality

List all the four equations written before. { & AB= DE & (I)& & BC= CD & (II)& & AC = AB + BC & (III)& & CE = DE + CD & (IV) & . According to the Substitution Property of Equality, equal values can replace each other in equations. Substitute Equations (I) and (II) into Equation (III). AC = DE + CD The right-hand side sum is the same as in Equation (IV). Then, substitute the left-hand side of Equation (IV) into this last equation. AC = CE

Apply the Definition of Congruence

If two segments have equal lengths, then the segments are congruent. AC = CE ⇓ AC≅CE It is concluded that AC is congruent to CE.

Once the proof is done, it can be compactly summarized in a paragraph. Given: & Cis a point onAE, Bis a point onAC, & Dis a point onCE, AB ≅ DE, & BC ≅ CD Prove: & AC ≅ CE Paragraph Proof: According to the definition of congruence, AB=DE and BC=CD. The Segment Addition Postulate states that AC=AB+BC and CE=CD+DE. By the Substitution Property of Equality and the Commutative Property of Addition, it follows that AC=CE. Since segments with equal lengths are congruent, this completes the proof that AC ≅ CE.

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