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 $\overline{A E},$ $B$ be a point on $\overline{A C},$ and $D$ be a point on $\overline{C E},$ such that $\overline{A B} ≅ \overline{D E}$ and $\overline{B C} ≅ \overline{C D} .$ Prove that $\overline{A C} ≅ \overline{C E} .$

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

Draw a Diagram

According to the given information, $C$ is a point on $\overline{A E} .$

Also, it is given that $B$ is a point on $\overline{A C}$ and $D$ is a point on $\overline{C E} .$

From the last piece of given information, $\overline{A B}$ is congruent to $\overline{D E}$ and $\overline{B C}$ is congruent to $\overline{C D} .$

Apply the Definition of Congruence

The given congruence statements imply that the congruent segments have equal lengths.

\overline{A B} ≅ \overline{D E} and \overline{B C} ≅ \overline{C D} ⇓ A B = D E and B C = C D

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.

A C C E = A B + B C = C D + D E

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.

C E = C D + D E ⇓ C E = D E + C D

Use the Substitution Property of Equality

List all the four equations written before.

⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ A B = D E B C = C D A C = A B + B C C E = D E + C D (I) (II) (III) (IV)

According to the Substitution Property of Equality, equal values can replace each other in equations. Substitute Equations (I) and (II) into Equation (III).

A C = D E + C D

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.

A C = C E

Apply the Definition of Congruence

If two segments have equal lengths, then the segments are congruent.

A C = C E ⇓ \overline{A C} ≅ \overline{C E}

It is concluded that

\overline{A C}

is congruent to

\overline{C E} .

A Line Segment With Two Congruent Subsegments

Once the proof is done, it can be compactly summarized in a paragraph.

Given : Prove : C is a point on \overline{A E}, B is a point on \overline{A C}, D is a point on \overline{C E}, \overline{A B} ≅ \overline{D E}, \overline{B C} ≅ \overline{C D} \overline{A C} ≅ \overline{C E}

Paragraph Proof: According to the definition of congruence,

A B = D E

and

B C = C D .

The Segment Addition Postulate states that

A C = A B + B C

and

C E = C D + D E .

By the Substitution Property of Equality and the Commutative Property of Addition, it follows that

A C = C E .

Since segments with equal lengths are congruent, this completes the proof that

\overline{A C} ≅ \overline{C E} .

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